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TU B.Sc. First Year Syllabus: Statistics
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TU B.Sc. First Year Statistics (Stat. 101) — Complete Syllabus Overview

Statistics is one of the major subjects offered under the Four-Year Bachelor of Science (B.Sc.) programme of Tribhuvan University’s Institute of Science and Technology (IoST), Kirtipur, Kathmandu. The First Year Statistics course, officially titled Fundamentals of Statistics (Course No. Stat. 101), is a 75-mark theory paper covering two groups: Group A — Descriptive Statistics and Group B — Probability and Probability Distributions. The course totals 150 lecture hours and is a pure theory paper with no separate practical paper at First Year level (practical statistics is introduced in later years).

The B.Sc. Statistics programme was upgraded from three years to four years in 2069 BS (2012 AD), and the current syllabus reflects the Revised Course of Study 2073 BS (2016). Statistics is offered as a Physical Group subject, typically combined with Mathematics and either Physics or Computer Science in First Year.

Subject Information at a Glance

  • Course Title: Fundamentals of Statistics
  • Course No.: Stat. 101
  • Nature of Course: Theory (no practical paper at First Year level)
  • Full Marks: 75 | Pass Marks: 35% (26.25)
  • Total Lecture Hours: 150
  • Year: First Year — Four Year B.Sc. Programme
  • Examination: Annual (end-of-year exam conducted by IoST, TU)
  • Conducting Institution: Institute of Science and Technology (IoST), Tribhuvan University
  • Curriculum Version: Revised Course of Study 2073 BS (2016)
  • Eligibility: Passed +2 or equivalent with minimum C grade in all subjects; must have studied Mathematics with 200 full marks
  • Total Institutions Offering: 2 institutions in Nepal (both in Kathmandu district) as of recent data

Why Statistics?

Statistics is the science of collecting, organizing, analysing, interpreting, and presenting data. In an era driven by data — where every field from medicine and economics to engineering, agriculture, and governance relies on quantitative evidence — Statistics has become one of the most broadly applicable and in-demand disciplines in the world. The statistician’s toolkit — probability theory, hypothesis testing, regression analysis, sampling design, and multivariate analysis — underpins virtually every empirical research methodology across the natural and social sciences.

In Nepal’s context, statistical expertise is directly needed in government planning (Central Bureau of Statistics — CBS, now National Statistics Office — NSO), public health surveillance (Health Management Information System — HMIS, epidemiological studies), banking and insurance (actuarial calculations, risk modelling), development research (poverty measurement, census analysis), agriculture (crop forecasting, field trial analysis), and the growing data analytics sector. The TU B.Sc. Statistics programme is designed to produce graduates with rigorous theoretical foundations and applied statistical competence, preparing them for careers across all quantitative sectors of Nepal’s economy.

Stat. 101 Theory Syllabus — Group A: Descriptive Statistics

Topic 1: Introduction to Statistics (5 teaching hours)

  • Meaning of Statistics: Statistics as a science; statistical data (plural meaning) vs. statistical method (singular meaning)
  • Importance of Statistics: role in knowledge generation, policy-making, and research across disciplines
  • Scope of Statistics: applications in Physical Sciences (Physics, Chemistry, Astronomy), Social Sciences (Economics, Sociology, Psychology), Management Sciences (business analytics, quality control, operations research), Information Technology (machine learning, data science, database analysis), Agriculture (crop trials, yield analysis, pest management), Insurance (actuarial science, premium calculation, life tables), Education (test development, educational research, performance evaluation), and Psychology (psychometrics, clinical trials, behavioral research)
  • Limitations of Statistics: does not deal with individual facts; results may be affected by bias; statistical laws are not exact; data can be misused; requires specialized knowledge for proper interpretation
  • Statistics and computer science: role of statistical programming (R, Python, SPSS, Stata) in modern data analysis

Topic 2: Population and Sample (8 teaching hours)

  • Types of characteristics: attributes (qualitative — nominal and ordinal) and variables (quantitative — discrete and continuous)
  • Scales of measurement: nominal scale (classification only — no ordering, e.g. blood group, gender), ordinal scale (ordering without equal intervals — e.g. education level, satisfaction rating), interval scale (equal intervals, no true zero — e.g. temperature in Celsius), ratio scale (equal intervals with true zero — e.g. weight, height, income)
  • Entities: statistical units of observation; types of entities
  • Types of data:
    • By source: primary data (collected directly for the study — field surveys, experiments, observation; sources in Nepal — CBS, DHM, DHM, MoH) and secondary data (already published — government reports, previous studies; their sources and uses)
    • By time dimension: cross-sectional data (collected at one point in time — snapshot of multiple units), time series data (repeated observations over time — GDP, inflation, rainfall records), failure time data (time until an event — survival data in medical research), panel data (combination — multiple units observed repeatedly over time)
  • Statistical population: definition (the complete set of units relevant to a study); finite population (countable — registered voters, students in a college) vs. infinite population (uncountable — gas molecules, future measurements); homogeneous population vs. heterogeneous population; target population vs. study population
  • Sample: definition; purpose of sampling (cost, time, feasibility, destructive testing)
  • Random sample vs. non-random (purposive) sample: definitions, implications for inference
  • Methods of sampling (description only, not mathematical theory):
    • Probability sampling: simple random sampling with replacement (SRSWR) — each draw is independent, same probability every draw; simple random sampling without replacement (SRSWOR) — each unit can be selected at most once; systematic sampling; stratified sampling (divide into homogeneous strata, sample from each); cluster sampling (divide into clusters, select entire clusters)
    • Non-probability sampling: convenience, purposive, snowball, quota sampling — uses and limitations

Topic 3: Presentation of Data (6 teaching hours)

  • Organization of data: data mining concept; editing (checking for errors, inconsistencies, missing values), coding (assigning numerical/categorical codes for analysis), data management and entry; assessing data quality (coverage, accuracy, completeness, timeliness)
  • Classification: organizing data into classes/groups; raw data and its classification; basis of classification (geographical, chronological, qualitative, quantitative)
  • Discrete frequency distribution: tally method; construction from raw data
  • Continuous frequency distribution: construction of class intervals — Sturge’s rule (k = 1 + 3.322 log₁₀ N); inclusive vs. exclusive method of classification; open-end classes; class boundaries, class marks, class width
  • Cumulative frequency distribution: less than and more than types; relative frequency distribution; percentage frequency distribution
  • Tabulation: one-way, two-way, and multi-way tables; rules of tabulation; construction of bivariate frequency distribution (joint frequency table for two variables)
  • Diagrammatic presentation: simple bar diagram, multiple bar diagram, sub-divided (component) bar diagram, pie chart — construction, interpretation, appropriate use (review of Grade 12 content with extension)
  • Graphical presentation:
    • Histogram: equal and unequal class widths; frequency density
    • Frequency polygon: construction from histogram; smoothed frequency curve
    • Ogive curves: less-than ogive and more-than ogive; use in determining median, quartiles graphically
    • Stem and leaf chart: construction and advantages over grouped frequency distribution
    • Range chart: control chart concept
    • Check sheet: tally-based data collection tool for quality control
    • Pareto diagram: bar chart + cumulative frequency line; 80-20 rule (Pareto principle); applications in quality management and public health priority-setting

Topic 4: Measures of Central Tendency and Dispersion (17 teaching hours — largest Group A topic)

Measures of Central Tendency

  • Concept of central tendency: the representative value of a dataset; properties of a good average
  • Arithmetic mean: simple and weighted arithmetic mean; mathematical properties (sum of deviations from mean = 0; sum of squared deviations is minimum about the mean — proof); trimmed mean; combined mean for k groups; effect of change of origin and scale; computation for grouped and ungrouped data
  • Median: definition; formula for grouped data (derivation of interpolation formula); graphical method using ogive; properties; effect of extreme values
  • Mode: definition; formula for grouped data (derivation); graphical method using histogram; properties; multimodal distributions; limitations
  • Harmonic mean: definition; weighted harmonic mean; uses (average rates, average speeds); computation
  • Geometric mean: definition; weighted geometric mean; uses (index numbers, growth rates); computation; combined geometric mean
  • Order relation: AM ≥ GM ≥ HM (with proof for n = 2 case — using AM-GM inequality)
  • Empirical relationship between mean, median, and mode: Karl Pearson’s empirical formula Mode ≈ 3 Median − 2 Mean (for moderately skewed distributions); interpretation
  • Choice of appropriate average: criteria — type of data, purpose of analysis, presence of extreme values; which average is best for which situation

Measures of Dispersion

  • Concept of dispersion: need for measuring variability; absolute vs. relative measures of dispersion
  • Range: definition; coefficient of range; merits and demerits; use in quality control (R chart)
  • Quartile deviation (semi-interquartile range): Q1, Q3; coefficient of quartile deviation; computation; merits and demerits
  • Mean deviation: definition for ungrouped and grouped data; mean deviation from mean and from median; minimality property — mean deviation is minimum when deviations are taken from the median (with proof); coefficient of mean deviation
  • Mean square deviation: definition; minimality property — mean square deviation is minimum when deviations are taken from the arithmetic mean (with proof)
  • Variance and standard deviation: definition (population and sample); mathematical properties of standard deviation (effect of adding a constant — origin; multiplying by a constant — scale; combined standard deviation formula); combined variance for 2 independent groups (derivation); generalization for n groups
  • Coefficient of variation (CV): definition (σ/μ × 100); interpretation; comparison of variability across datasets with different units or different means
  • Theoretical problems of measures of dispersion
  • Five number summary: minimum, Q1, median, Q3, maximum; use in exploratory data analysis
  • Box plot (box-and-whisker plot): construction and interpretation; identification of outliers; comparison of distributions
  • Normal probability plot: concept; use in checking normality assumption
  • Lorenz curve: graphical representation of inequality; construction — cumulative proportion of population vs. cumulative proportion of income/wealth
  • Gini coefficient: numerical measure of inequality; interpretation (0 = perfect equality, 1 = perfect inequality); computation from Lorenz curve

Topic 5: Moments, Skewness and Kurtosis (12 teaching hours)

Moments

  • Raw moments (mᵣ) about origin: definition mᵣ = Σf xʳ / N for grouped and Σxʳ/n for ungrouped data; first four raw moments
  • Moments about an arbitrary constant: mᵣ(a) = Σf(x−a)ʳ / N; computation; use in simplifying calculation of central moments
  • Central moments (μᵣ): definition μᵣ = Σf(x−x̄)ʳ / N; first four central moments (μ₁ = 0; μ₂ = variance; μ₃ and μ₄ for skewness and kurtosis)
  • Effect of change of origin and scale on moments: mᵣ(Y) when Y = (X−a)/h
  • Relations between central moments and raw moments (up to 4th order): μ₁ = 0; μ₂ = m₂ − m₁²; μ₃ = m₃ − 3m₂m₁ + 2m₁³; μ₄ = m₄ − 4m₃m₁ + 6m₂m₁² − 3m₁⁴

Skewness

  • Concept of skewness: asymmetry of a frequency distribution; positive skewness (right-skewed — long right tail; mean > median > mode), negative skewness (left-skewed — long left tail; mean < median < mode), symmetric distribution (mean = median = mode)
  • Bowley’s coefficient of skewness: based on quartiles — SK = (Q3 + Q1 − 2 Median)/(Q3 − Q1); range [−1, 1]; computation and interpretation; interpretation using box plot (quartile skewness)
  • Karl Pearson’s coefficient of skewness: SK = (Mean − Mode)/SD or SK = 3(Mean − Median)/SD (using empirical relation); range approximately [−3, +3]
  • Measures of skewness based on moments: β₁ = μ₃²/μ₂³ (always ≥ 0; = 0 for symmetric); γ₁ = √β₁ (takes sign of μ₃); interpretation

Kurtosis

  • Concept of kurtosis: peakedness (or flatness) of a frequency distribution relative to normal distribution
  • Types: leptokurtic (β₂ > 3 — more peaked than normal, heavy tails), mesokurtic (β₂ = 3 — exactly normal), platykurtic (β₂ < 3 — flatter than normal, lighter tails)
  • Measures of kurtosis using partition values: based on deciles and percentiles — percentile kurtosis
  • Measures of kurtosis based on moments: β₂ = μ₄/μ₂² (Pearson’s kurtosis); γ₂ = β₂ − 3 = μ₄/μ₂² − 3 (excess kurtosis — = 0 for normal distribution); interpretation

Topic 6: Introduction to Correlation (12 teaching hours)

  • Bivariate data: paired observations (x, y); bivariate frequency distribution (joint frequency table for two variables); marginal distributions
  • Correlation between two variables: concept — the degree and direction of linear relationship; positive correlation (both variables increase together), negative correlation (one increases as the other decreases), zero correlation (no linear relationship); spurious correlation
  • Scatter diagram: construction; interpretation of direction and strength of relationship; limitation (visual only, no numerical measure)
  • Covariance between two variables: definition Cov(X,Y) = Σ(xᵢ−x̄)(yᵢ−ȳ)/n; computation for ungrouped and grouped data; effect of change of origin and scale on covariance; limitation (scale-dependent)
  • Karl Pearson’s coefficient of correlation (r): definition r = Cov(X,Y)/(σₓ σᵧ); computation for grouped and ungrouped data; assumptions (linearity, bivariate normality, absence of outliers)
  • Properties of r (with proof):
    • (i) −1 ≤ r ≤ 1 (range property — proof using Cauchy-Schwarz inequality or AM-GM argument)
    • (ii) Effect of change of origin and scale: r is independent of change of origin and scale (r is dimensionless and scale-free)
  • Interpretation: r = +1 (perfect positive linear), r = −1 (perfect negative linear), r = 0 (no linear relationship); r near ±1 (strong), r near 0 (weak)
  • Spearman’s rank correlation coefficient: definition ρ = 1 − 6Σd²/[n(n²−1)] where d = difference in ranks; computation for untied and tied ranks (correction factor for ties); use when data are ordinal or when normality assumption fails; comparison with Pearson’s r

Topic 7: Regression Analysis (15 teaching hours)

  • Concept of regression: prediction of one variable from another; distinction from correlation (correlation measures association; regression measures the form and strength of the relationship for prediction)
  • Lines of regression: two lines — regression of Y on X (used to predict Y given X) and regression of X on Y (used to predict X given Y); concept of why two lines exist (different minimization criteria)
  • Fitting by the method of least squares: principle (minimize sum of squared residuals); derivation of normal equations for regression of Y on X: ΣY = na + bΣX; ΣXY = aΣX + bΣX²; solution for slope b = (nΣXY − ΣXΣY)/(nΣX² − (ΣX)²) and intercept a = Ȳ − bX̄
  • Interpretation: slope b (change in Y per unit change in X); intercept a (predicted Y when X = 0); concept of linearity assumption
  • Regression coefficients:
    • bᵧₓ (regression coefficient of Y on X): bᵧₓ = r·(σᵧ/σₓ)
    • bₓᵧ (regression coefficient of X on Y): bₓᵧ = r·(σₓ/σᵧ)
    • Properties (with proof):
      • (i) bᵧₓ · bₓᵧ = r² (geometric mean of regression coefficients = r)
      • (ii) bᵧₓ · bₓᵧ ≤ 1 (since r² ≤ 1)
      • (iii) bᵧₓ = r·σᵧ/σₓ, bₓᵧ = r·σₓ/σᵧ (explicit formula with proof)
      • (iv) Effect of change of origin and scale on regression coefficients
      • (v) Angle between the two lines of regression: tan θ = [(1 − r²)/r] · [σₓσᵧ/(σₓ² + σᵧ²)]; when r = ±1, lines coincide; when r = 0, lines are perpendicular (at right angles)
  • Both regression lines pass through (X̄, Ȳ)
  • Mean residual sum of squares (Mean Squared Error — MSE): definition; computation; interpretation as average prediction error
  • Residual plot: plot of residuals (ŷᵢ − yᵢ) against fitted values (ŷᵢ) or against X; interpretation — random scatter indicates good fit; patterns (curve, fan shape) indicate assumption violations
  • Partitioning of total variation: Total Sum of Squares (SST) = Regression Sum of Squares (SSR) + Error Sum of Squares (SSE); explained vs. unexplained variation
  • Coefficient of determination (r²): proportion of total variation in Y explained by X; r² = SSR/SST; interpretation (r² = 0.80 means 80% of variability in Y is explained by the linear relationship with X); relationship to Pearson’s r
  • Concept of multiple regression: extending to two or more predictors (Y = β₀ + β₁X₁ + β₂X₂ + … + βₖXₖ + ε); interpretation of partial regression coefficients; applications in Nepal — economic growth modelling, health outcome prediction, agricultural yield modelling

Stat. 101 Theory Syllabus — Group B: Probability and Distributions

Topic 8: Introduction to Probability (20 teaching hours — largest single topic)

Review of Set Theory

  • Set operations: union (A ∪ B), intersection (A ∩ B), complement (Aᶜ), difference (A − B), symmetric difference
  • De Morgan’s laws: (A ∪ B)ᶜ = Aᶜ ∩ Bᶜ; (A ∩ B)ᶜ = Aᶜ ∪ Bᶜ; proofs using Venn diagrams and element-wise arguments
  • Useful set identities for probability: distributive laws, inclusion-exclusion principle

Concepts in Probability

  • Deterministic vs. random experiments: definition and distinction; examples
  • Definitions: trial (single performance of an experiment), event (any subset of sample space), outcome (a single result), sample space Ω (set of all possible outcomes)
  • Types of events: equally likely events; mutually exclusive events (A ∩ B = ∅); exhaustive events (A₁ ∪ A₂ ∪ … ∪ Aₙ = Ω); complementary events; independent vs. dependent events; sure event (Ω); impossible event (∅)

Definitions of Probability

  • Mathematical (classical) definition: P(A) = n(A)/n(Ω) where all outcomes are equally likely; merits (simple, widely applicable); demerits (requires equally likely outcomes, not applicable to continuous sample spaces, circular reasoning in defining “equally likely”)
  • Statistical (relative frequency) definition: P(A) = lim(n→∞) f/n where f = frequency of event A in n trials; merits (empirical, no assumption of equal likelihood); demerits (limit may not exist, idealized — infinite trials not possible in practice)
  • Subjective probability: personal degree of belief; used in Bayesian inference; merits and demerits
  • Axiomatic definition (Kolmogorov 1933): three axioms — (1) P(A) ≥ 0 for all A; (2) P(Ω) = 1; (3) P(A ∪ B) = P(A) + P(B) for mutually exclusive A, B (countable additivity); this definition unifies all others and is the modern foundation

Combinatorial Analysis and Probability

  • Fundamental counting principle: multiplication rule (if event 1 can occur in m ways and event 2 in n ways, total = m × n)
  • Permutations: nPr = n!/(n−r)!; permutations with repetition; circular permutations; permutations with identical objects
  • Combinations: nCr = n!/[r!(n−r)!]; properties (nCr = nCₙ₋ᵣ); Pascal’s triangle; binomial coefficient
  • Combinatorial probability examples: drawing cards, rolling dice, random selection problems; classical probability computations using combinations

Algebra of Events and Probability Theorems

  • Properties of probability: P(∅) = 0; P(Aᶜ) = 1 − P(A); if A ⊆ B then P(A) ≤ P(B); 0 ≤ P(A) ≤ 1
  • Additive theorem (inclusion-exclusion): P(A ∪ B) = P(A) + P(B) − P(A ∩ B); generalization for three events; extension to n events
  • Boole’s inequality (union bound): P(A₁ ∪ A₂ ∪ … ∪ Aₙ) ≤ P(A₁) + P(A₂) + … + P(Aₙ) (proof and application in bounding probabilities)
  • Multiplicative theorem: P(A ∩ B) = P(A) · P(B|A) = P(B) · P(A|B); extension to three events

Geometric Probability and Bertrand’s Paradox

  • Geometric probability: probability as ratio of measure (length, area, volume) in continuous sample spaces; problems on geometric probability (points chosen at random on a line, in a region)
  • Bertrand’s paradox: three different methods of choosing a “random chord” in a circle give three different probabilities — illustrating that probability depends on the precise definition of “random”; philosophical implications for the classical definition

Conditional Probability and Independence

  • Conditional probability: P(A|B) = P(A ∩ B)/P(B) for P(B) > 0; interpretation — updated probability of A given knowledge that B occurred
  • Pairwise independence vs. mutual independence: pairwise — each pair satisfies P(Aᵢ ∩ Aⱼ) = P(Aᵢ)·P(Aⱼ); mutual independence — all subsets satisfy the multiplicative rule; note that pairwise independence does NOT imply mutual independence (counterexample)
  • Total probability theorem: P(A) = Σᵢ P(A|Bᵢ) · P(Bᵢ) where {B₁, B₂, …, Bₙ} is a partition of Ω; proof and applications

Bayes’ Theorem and Diagnostic Test Applications

  • Bayes’ theorem: P(Bᵢ|A) = P(A|Bᵢ)·P(Bᵢ) / Σⱼ P(A|Bⱼ)·P(Bⱼ); prior probability P(Bᵢ) and posterior probability P(Bᵢ|A); derivation
  • Applications of Bayes’ theorem: disease diagnosis, quality control (proportion of defective items from different machines), forensic evidence, spam filtering
  • Medical diagnostic test applications (direct relevance to Nepal’s health statistics context):
    • Sensitivity: P(test positive | disease present) — true positive rate
    • Specificity: P(test negative | disease absent) — true negative rate
    • Predictive Value Positive (PVP): P(disease present | test positive) — computed using Bayes’ theorem with disease prevalence as prior; depends on prevalence (a positive TB test in a high-prevalence country like Nepal has different PVP than in a low-prevalence country)
    • Predictive Value Negative (PVN): P(disease absent | test negative)
    • Numerical examples using HIV, TB, pregnancy test scenarios

Topic 9: Random Variables (10 teaching hours)

  • Concept of a random variable: a function X: Ω → ℝ that assigns a real number to each outcome in the sample space; distinction from an algebraic variable (random variable has a probability distribution)
  • Types of random variables:
    • Discrete random variable: takes countable values (finite or countably infinite); examples (number of heads in 5 coin flips; number of TB cases per week in a district — relevant Nepal public health example)
    • Continuous random variable: takes any value in an interval (uncountably many values); examples (height, weight, blood pressure, waiting time, rainfall amount)
  • Probability distribution of a random variable:
    • Probability mass function (PMF) for discrete: f(x) = P(X = x); properties — f(x) ≥ 0; Σf(x) = 1
    • Probability density function (PDF) for continuous: f(x) satisfying f(x) ≥ 0; ∫f(x)dx = 1; P(a ≤ X ≤ b) = ∫ₐᵇf(x)dx
  • Distribution function (Cumulative Distribution Function — CDF): F(x) = P(X ≤ x); properties — non-decreasing; right-continuous; F(−∞) = 0; F(+∞) = 1; for discrete: step function; for continuous: F(x) = ∫₋∞ˣ f(t)dt; P(a < X ≤ b) = F(b) − F(a)
  • Functions of random variables: if Y = g(X), finding distribution of Y from distribution of X; linear transformations (Y = aX + b) for both discrete and continuous; nonlinear transformations (Y = X², Y = |X|) — concept and examples; applications in standardization (Z = (X−μ)/σ)

Topic 10: Theory of Mathematical Expectation (15 teaching hours)

  • Mathematical expectation of a random variable: E(X) = Σx·f(x) (discrete); E(X) = ∫x·f(x)dx (continuous); interpretation as the weighted average of all possible values; existence condition for E(X)
  • Expectation of a function: E[g(X)] = Σg(x)·f(x) (discrete); E[g(X)] = ∫g(x)·f(x)dx (continuous)
  • Properties of mathematical expectation (with proofs):
    • E(c) = c (expectation of a constant)
    • E(aX + b) = aE(X) + b (linearity)
    • E(aX + bY) = aE(X) + bE(Y) (linearity for sum — no independence required)
    • E(XY) = E(X)·E(Y) if X and Y are independent
  • Addition theorem of expectation: E(X₁ + X₂ + … + Xₙ) = E(X₁) + E(X₂) + … + E(Xₙ) (regardless of dependence)
  • Multiplicative theorem of expectation: E(X₁ · X₂ · … · Xₙ) = E(X₁) · E(X₂) · … · E(Xₙ) if X₁, X₂, …, Xₙ are mutually independent
  • Variance: Var(X) = E[(X − μ)²] = E(X²) − [E(X)]² (computational formula); interpretation (measure of spread around the mean); Var(aX + b) = a²Var(X) (effect of linear transformation)
  • Covariance: Cov(X,Y) = E[(X−μₓ)(Y−μᵧ)] = E(XY) − E(X)E(Y); interpretation; Cov(X,Y) = 0 if X and Y are independent (converse not always true)
  • Correlation: ρ(X,Y) = Cov(X,Y)/[σₓ·σᵧ]; range [−1, 1]
  • Variance of a linear combination: Var(aX + bY) = a²Var(X) + b²Var(Y) + 2ab·Cov(X,Y); for independent: Var(aX + bY) = a²Var(X) + b²Var(Y); generalization for n variables
  • Conditional expectation: E(X|Y = y) = Σx·f(x|y) (discrete); E(E(X|Y)) = E(X) (law of total expectation)
  • Conditional variance: Var(X|Y) = E(X²|Y) − [E(X|Y)]²; law of total variance: Var(X) = E[Var(X|Y)] + Var[E(X|Y)]
  • Moments of random variables: raw moments μᵣ’ = E(Xʳ); central moments μᵣ = E[(X−μ)ʳ]; using moments to find measures of location (μ₁’ = mean), dispersion (μ₂ = variance), skewness (γ₁ = μ₃/σ³), kurtosis (β₂ = μ₄/σ⁴)
  • Generating functions:
    • Moment generating function (MGF): Mₓ(t) = E(eᵗˣ); existence; properties — moments obtained as derivatives: E(Xʳ) = Mₓ⁽ʳ⁾(0); MGF uniquely determines the distribution; MGF of sum of independent random variables = product of individual MGFs
    • Probability generating function (PGF): G(t) = E(tˣ) for non-negative integer valued X; relation to probabilities and moments
    • Cumulant generating function (CGF): Kₓ(t) = ln[Mₓ(t)]; cumulants κ₁ = μ, κ₂ = σ²; advantage — cumulants are additive for independent variables
    • Characteristic function: φₓ(t) = E(eⁱᵗˣ) (using complex exponential); always exists (unlike MGF); properties; use in proving Central Limit Theorem

Topic 11: Probability Distributions (30 teaching hours — largest topic in the syllabus)

Discrete Probability Distributions

  • Bernoulli trial: single binary experiment with success probability p and failure probability q = 1−p; Bernoulli distribution — PMF f(x) = pˣ(1−p)¹⁻ˣ for x = 0, 1; E(X) = p; Var(X) = pq
  • Binomial distribution:
    • Setting: n independent Bernoulli trials, each with success probability p; X = number of successes
    • PMF: P(X = x) = ₙCₓ pˣ qⁿ⁻ˣ for x = 0, 1, 2, …, n
    • Distribution function: F(x) = Σᵢ₌₀ˣ ₙCᵢ pⁱ qⁿ⁻ⁱ
    • MGF: Mₓ(t) = (q + peᵗ)ⁿ; characteristic function
    • Moments: E(X) = np; Var(X) = npq; μ₃ = npq(q−p); μ₄ = npq[1 + 3(n−2)pq]; β₁ = (q−p)²/(npq); β₂ = 3 + (1−6pq)/(npq)
    • Properties: additive property (sum of independent binomials with same p is binomial); recurrence relation for probabilities; mode
    • Distribution fitting: computing theoretical frequencies using binomial probabilities; chi-square goodness of fit test (concept for fitting)
    • Applications in Nepal context: number of malaria-positive patients in a sample from an endemic district; number of defective tablets in a pharmaceutical batch; number of successful loan repayments in a portfolio
  • Poisson distribution:
    • Setting: count of rare events in a fixed interval of time or space; derived as limit of binomial when n→∞, p→0, np = λ (constant)
    • PMF: P(X = x) = e⁻λ λˣ/x! for x = 0, 1, 2, …
    • Distribution function (series form)
    • MGF: Mₓ(t) = exp[λ(eᵗ−1)]; characteristic function
    • Moments: E(X) = λ; Var(X) = λ (mean = variance — key property for testing); μ₃ = λ; μ₄ = λ + 3λ²; β₁ = 1/λ; β₂ = 3 + 1/λ
    • Properties: additive property; recurrence relation P(X = x) = (λ/x) P(X = x−1); mode; Poisson approximation to binomial (when n large, p small, np = λ moderate — rule of thumb: n ≥ 20, p ≤ 0.05)
    • Distribution fitting: computing theoretical frequencies; goodness of fit
    • Applications in Nepal context: number of traffic accidents per day on Prithvi Highway; number of disease cases per week in a district; number of incoming calls per minute to a health helpline; number of floods per monsoon season exceeding a threshold

Continuous Probability Distributions

  • Rectangular (Uniform) distribution:
    • Setting: equally likely outcomes over interval [a, b]
    • PDF: f(x) = 1/(b−a) for a ≤ x ≤ b; 0 otherwise
    • CDF: F(x) = (x−a)/(b−a) for a ≤ x ≤ b
    • MGF: Mₓ(t) = (eᵗᵇ − eᵗᵃ)/[t(b−a)]; characteristic function
    • Moments: E(X) = (a+b)/2; Var(X) = (b−a)²/12; β₁ = 0 (symmetric); β₂ = 9/5 (platykurtic relative to normal)
    • Properties: use in simulation (uniform random number generation as the foundation of all simulation); use in establishing probability bounds
  • Normal distribution:
    • Setting: continuous symmetric distribution; fundamental due to Central Limit Theorem
    • PDF: f(x) = (1/σ√(2π)) exp[−(x−μ)²/(2σ²)] for −∞ < x < ∞; parameters μ (mean) and σ² (variance)
    • Bell-shaped, symmetric about μ; inflection points at μ ± σ; asymptotic to x-axis
    • CDF: Φ(x) — no closed form; use of standard normal table
    • MGF: Mₓ(t) = exp(μt + σ²t²/2); characteristic function
    • Moments: E(X) = μ; Var(X) = σ²; all odd central moments = 0 (symmetry); μ₄ = 3σ⁴; β₁ = 0; β₂ = 3 (mesokurtic by definition)
    • Properties:
      • Linear combination: if X₁ ~ N(μ₁, σ₁²) and X₂ ~ N(μ₂, σ₂²) independent, then aX₁ + bX₂ ~ N(aμ₁ + bμ₂, a²σ₁² + b²σ₂²)
      • Additive property for independent normals
      • Symmetry: f(μ+x) = f(μ−x); P(μ−kσ < X < μ+kσ) — the 68-95-99.7 rule (1σ: 68.27%; 2σ: 95.45%; 3σ: 99.73%)
    • Standard normal distribution: Z = (X−μ)/σ ~ N(0,1); use of Z-tables; computation of probabilities for any normal
    • Areas under the standard normal curve: P(0 < Z < z), P(Z > z), P(Z < z), P(−z < Z < z) — using standard normal table
    • Normal approximation to binomial: when n large and p not extreme; conditions np ≥ 5 and nq ≥ 5; continuity correction factor (P(X = k) ≈ P(k−0.5 < X < k+0.5))
    • Normal approximation to Poisson: when λ is large (λ ≥ 30 approximately); Z = (X−λ)/√λ
    • Distribution fitting: computing theoretical normal frequencies from mean and standard deviation; Q-Q plot for assessing normality
    • Applications in Nepal context: distribution of heights, weights, blood pressure readings in population surveys (Nepal Demographic and Health Survey — NDHS); test scores; agricultural yield measurements

The Four-Year TU B.Sc. Statistics Programme Structure

  • First Year: Stat. 101 Fundamentals of Statistics + Mathematics I (MAT 101 Calculus) + Mathematics II (MAT 102 Analytical Geometry & Vector Analysis) + Scientific Communication
  • Second Year: Stat. 201 Applied Statistics (or Probability and Inference I) + additional mathematics/physics
  • Third Year: Stat. 301 Probability and Inference II + Stat. 302 Economic Statistics (or Statistical Methods, Sampling Theory and Time Series)
  • Fourth Year: Advanced Applied Statistics, Regression Analysis, Time Series, Non-parametric Statistics, Design of Experiments, Research Project

Recommended Textbooks and References

Primary References (as listed in the official Stat. 101 syllabus)

  • Miller, I. and Freund, J. (2007). Modern Elementary Statistics. Pearson Publishers
  • Snedecor, G.W. and Cochran, W.G. (1980). Statistical Methods. Oxford and IBH Publishers
  • Gupta, S.C. and Kapoor, V.K. (2012). Fundamentals of Mathematical Statistics. Sultan Chand and Sons, New Delhi (widely used across South Asian statistics programmes)
  • Shrestha, H.B. Statistics and Probability: Concepts and Techniques. EKTA Books, Kathmandu (Nepal-specific, widely available)
  • Sthapit, A., Yadav, R. and Khanal, S. (2012). Fundamentals of Statistics. Asmita Publication, Kathmandu, Nepal (Nepal-specific primary reference — directly aligned with TU syllabus)
  • Sukubhattu, N.P. (2063 BS). Probability Theory and Statistical Methods. 2nd Edition. Asmita Publications, Kathmandu (Nepal-specific, strong on probability theory)
  • Daniel, W.W. (2000). Biostatistics: A Foundation for Analysis in the Health Sciences. 7th Edition. John Wiley and Sons (applied statistics with medical and public health examples)

Difference Between NEB Grade 12 and TU B.Sc. First Year Statistics

NEB Grade 12 Mathematics (Basic Mathematics Grade XII) covers statistics through correlation and regression (Karl Pearson’s coefficient, regression lines), and probability through conditional probability and Bayes’ theorem at an introductory computational level. TU B.Sc. First Year Statistics (Stat. 101) covers all of this — but at a fundamentally deeper level of mathematical rigour and with significant additional content. Measures of central tendency and dispersion are treated with formal proofs (minimality properties, combined variance derivations, moment relations). Correlation and regression include residual analysis, coefficient of determination, and concept of multiple regression. The probability section covers the axiomatic definition (Kolmogorov), geometric probability and Bertrand’s paradox, pairwise vs. mutual independence, and Bayes’ theorem applied to diagnostic test sensitivity and specificity — none of which appear in the NEB curriculum. Random variables, mathematical expectation with moment generating functions, cumulant generating functions, characteristic functions, and the theory of Binomial, Poisson, Uniform, and Normal distributions (each with full derivation of moments, MGF, and distribution fitting) are entirely new university-level content with no NEB equivalent.

Career Opportunities for TU B.Sc. Statistics Graduates

  • Government statistical agencies: National Statistics Office (NSO — formerly CBS), Nepal Rastra Bank (NRB), Ministry of Finance, Ministry of Health and Population — data collection, survey design, analysis, and publication of national statistics
  • Health and demographic research: Nepal Demographic and Health Survey (NDHS), Health Management Information System (HMIS), epidemiological research at NHRC, WHO Nepal, UNICEF — biostatistician, data analyst, survey statistician
  • Banking and insurance: Nepal Insurance Authority, commercial banks, development banks — actuarial assistant, risk analyst, credit modelling analyst
  • Development and research organizations: World Bank Nepal, ADB, UNDP Nepal, ICIMOD — quantitative research officer, monitoring and evaluation specialist, data scientist
  • Academic and teaching: statistics lecturer at higher secondary and university level (requires M.Sc. and TU/TSC clearance)
  • Data science and analytics: Nepal’s growing IT sector increasingly demands data scientists and analysts with strong statistical foundations — R, Python, SQL skills combined with statistics from TU B.Sc. open careers in tech companies, fintech, and e-commerce
  • Pharmaceutical and clinical research: statistical analysis of clinical trial data, pharmacovigilance, medical device testing — positions in Nepal’s expanding pharmaceutical sector and international CROs

Tips to Succeed in TU B.Sc. First Year Statistics

  • Group A is computational — practice problems daily: Measures of central tendency, dispersion, moments, correlation, and regression are all tested through numerical problems requiring accurate computation. Allocate at least 60% of your study time to working through problems, not reading theory. Create a formula card for all formulas (mean, variance, combined variance, regression coefficients, moment relations) and practice computing them for both grouped and ungrouped data until the procedure becomes automatic.
  • Group B requires conceptual depth alongside calculation: Probability and distributions require both conceptual understanding and computational ability. The proofs (Boole’s inequality, properties of r, regression coefficient properties, variance of a linear combination) are directly examined in theory questions. Learn each proof step by step — not just the final result.
  • Master Bayes’ theorem with the medical diagnostic framework: Bayes’ theorem combined with sensitivity, specificity, PPV, and NPV calculations is one of the most frequently examined topics in Group B. Practice several numerical problems — particularly ones involving low-prevalence diseases where the PPV is surprisingly low despite high sensitivity (base rate fallacy) — as this is a conceptual trap that examiners consistently test.
  • Understand the Normal distribution deeply: The Normal distribution (Topic 11) carries significant marks and appears in multiple forms — finding areas using Z-tables, normal approximation to binomial/Poisson, distribution fitting, and assessment of normality. Build a systematic procedure: always standardize first (Z = (X−μ)/σ), draw the normal curve, shade the region, and use the Z-table.
  • Use Nepali-authored textbooks as primary references: Sthapit, Yadav, and Khanal’s Fundamentals of Statistics (Asmita Publication) and Shrestha’s Statistics and Probability (EKTA Books) are both directly aligned with the TU Stat. 101 syllabus and widely available across Nepal. Sukubhattu’s Probability Theory and Statistical Methods is the strongest Nepali reference for Group B. These should be your first resources before reaching for Gupta & Kapoor or Miller & Freund.
  • The Gini coefficient and Lorenz curve are frequently examined: Despite appearing only once in Topic 4 (Measures of Dispersion), the Lorenz curve and Gini coefficient appear regularly in examination papers as they combine graphical and computational skills. Practice constructing a Lorenz curve from raw income distribution data and computing the Gini coefficient.
  • Solve the last 5 years of TU B.Sc. Statistics First Year past papers: The examination pattern is highly consistent — particular emphasis on Topic 4 (Measures of Central Tendency and Dispersion — 17 hours), Topic 7 (Regression Analysis — 15 hours), Topic 8 (Introduction to Probability — 20 hours), and Topic 11 (Probability Distributions — 30 hours). Past papers from EducateNepal (educatenepal.com) reveal exactly which sub-topics appear most frequently.

Frequently Asked Questions (FAQs)

What is the official course code and title for TU B.Sc. First Year Statistics?

The course code is Stat. 101 and the official course title is Fundamentals of Statistics. It carries 75 full marks with a pass mark of 35% (approximately 26.25). The course is a pure theory paper — there is no separate practical paper at First Year level.

How is Stat. 101 structured and how many teaching hours does each group have?

Stat. 101 has two groups across 11 topics and 150 teaching hours. Group A (Descriptive Statistics) covers Topics 1–7 with approximately 75 teaching hours: Introduction to Statistics (5 hrs), Population and Sample (8 hrs), Presentation of Data (6 hrs), Measures of Central Tendency and Dispersion (17 hrs), Moments, Skewness and Kurtosis (12 hrs), Introduction to Correlation (12 hrs), and Regression Analysis (15 hrs). Group B (Probability and Distributions) covers Topics 8–11 with approximately 75 teaching hours: Introduction to Probability (20 hrs), Random Variables (10 hrs), Theory of Mathematical Expectation (15 hrs), and Probability Distributions (30 hrs).

What is the eligibility for B.Sc. Statistics at TU?

Any student who has completed +2 Science or equivalent with a minimum C grade in all subjects and has studied Mathematics with 200 full marks at Grade 11 and 12 is eligible to apply. The Mathematics requirement distinguishes Statistics from biological group subjects — students who did not study Mathematics at full marks may not be directly eligible.

How many institutions offer B.Sc. Statistics under TU in Nepal?

Currently, approximately 2 institutions offer B.Sc. Statistics under Tribhuvan University in Nepal, both located in Kathmandu district. This is significantly fewer than Biology, Chemistry, or Physics — making B.Sc. Statistics a specialized programme with relatively limited institutional availability but correspondingly strong career differentiation for graduates.

Is there a practical paper in TU B.Sc. First Year Statistics?

No. Unlike Chemistry, Botany, Zoology, Physics, and Microbiology, First Year Statistics (Stat. 101) is a pure theory paper with no separate practical paper. Statistical computing (using R, SPSS, or similar software) and applied statistical practicals are introduced in later years.

Which textbooks are officially recommended for Stat. 101?

The official Stat. 101 syllabus recommends: Sthapit, Yadav, and Khanal’s Fundamentals of Statistics (Asmita Publication, Kathmandu), Shrestha’s Statistics and Probability: Concepts and Techniques (EKTA Books), Sukubhattu’s Probability Theory and Statistical Methods (Asmita Publications), Gupta and Kapoor’s Fundamentals of Mathematical Statistics (Sultan Chand and Sons), and Miller and Freund’s Modern Elementary Statistics (Pearson).

Where can I find Statistics notes and past papers for TU B.Sc. First Year?

You can find study resources on nneducation.com. The official Stat. 101 syllabus PDF is also embedded at the top of this page. Past papers for Fundamentals of Statistics (Stat. 101) are available through EducateNepal (educatenepal.com) and the IoST TU official website (tuiost.edu.np).

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