Engineering Mathematics
Linear Algebra Calculus Differential Equations Probability & Statistics Numerical Methods
Module 2 · GATE Engineering Mathematics

Probability & Statistics

Probability distributions, mean, variance, hypothesis testing, regression.
Probability · Distributions · Statistics · Regression
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Learning Objectives

  • Understand the fundamental concepts of Probability & Statistics
  • Apply key formulas and techniques to solve problems
  • Practice with exam-level questions to build speed and accuracy

Key Concepts

What is Probability & Statistics?

Probability and Statistics form a core part of engineering mathematics. This topic covers probability theory, random variables, probability distributions, sampling, estimation, hypothesis testing, and regression analysis. GATE weightage is approximately 5-8%.

Key Concepts

1. Probability Basics: P(E) = favourable/total equally likely outcomes. Additive rule: P(A∪B) = P(A)+P(B)-P(A∩B). Conditional: P(A|B) = P(A∩B)/P(B). Independence: P(A∩B) = P(A)P(B). Bayes theorem for inverse probability.

2. Random Variables: Discrete — probability mass function (PMF). Continuous — probability density function (PDF). Cumulative distribution function (CDF): F(x) = P(X ≤ x). Expectation: E[X] = ∑x·P(x) or ∫x·f(x)dx. Variance: Var(X) = E[X²] - (E[X])².

3. Standard Distributions: Binomial: X~B(n,p), P(X=r)=ⁿCᵣ pʳqⁿ⁻ʳ, E[X]=np, Var(X)=npq. Poisson: X~P(λ), P(X=r)=e⁻λ λʳ/r!, E[X]=λ, Var(X)=λ. Normal: X~N(μ,σ²), bell-shaped, symmetry about μ, standard normal Z = (X-μ)/σ ~ N(0,1). Uniform: X~U(a,b), f(x)=1/(b-a) for a≤x≤b.

4. Sampling & Estimation: Central Limit Theorem: sample mean ~ N(μ, σ²/n) for large n. Confidence intervals: CI = X̄ ± Z(α/2)×σ/√n. Point estimate vs. interval estimate.

5. Correlation & Regression: Correlation coefficient r measures linear relationship (-1 ≤ r ≤ 1). Regression line: y = a + bx, where b = Cov(X,Y)/Var(X), a = ӯ - bx̄.

Solved Examples

Example 1 (Expectation & Variance)
A random variable X has PMF: P(X=0)=0.2, P(X=1)=0.5, P(X=2)=0.3. Find E[X] and Var(X).

a) E=1.1, Var=0.49 b) E=1.2, Var=0.56 c) E=1.1, Var=0.69 d) E=1.0, Var=0.5
Solution: a) E=1.1, Var=0.49. E[X] = 0×0.2+1×0.5+2×0.3 = 0+0.5+0.6 = 1.1. E[X²] = 0²×0.2+1²×0.5+2²×0.3 = 0+0.5+1.2 = 1.7. Var = 1.7 - (1.1)² = 1.7 - 1.21 = 0.49.
Example 2 (Normal Distribution)
If X~N(50, 16), find P(46 < X < 54). Given P(Z < 1) = 0.8413.

a) 0.6826 b) 0.9544 c) 0.4972 d) 0.8413
Solution: a) 0.6826. μ=50, σ=4. Z₁ = (46-50)/4 = -1, Z₂ = (54-50)/4 = 1. P(-1
Example 3 (Binomial Distribution)
In a manufacturing process, 5% of items are defective. What is the probability that in a sample of 10 items, at most one is defective?

a) 0.9138 b) 0.5987 c) 0.9139 d) 0.9885
Solution: a) 0.9138. n=10, p=0.05, q=0.95. P(X≤1) = P(0) + P(1). P(0) = ¹⁰C₀(0.05)⁰(0.95)¹⁰ = 0.95¹⁰ = 0.5987. P(1) = ¹⁰C₁(0.05)¹(0.95)⁹ = 10×0.05×0.95⁹ = 0.5×0.6302 = 0.3151. Total = 0.5987+0.3151 = 0.9138.

Shortcut Techniques

  • For binomial probability, use recurrence: P(r+1) = P(r)×(n-r)/(r+1)×p/q to avoid recomputing from scratch.
  • Normal distribution problems use symmetry: P(-k
  • Variance shortcut: Var(X) = E[X²]-(E[X])² — this is almost always easier than the definition.
  • For Poisson distribution with large n and small p (n>50, p<0.1), approximate binomial as Poisson with λ=np.
  • Central Limit Theorem: for any population, sample mean distribution approaches normal as n increases (n≥30 sufficient).
Pro Tip
In GATE, probability and statistics questions are scoring if you memorise the standard distributions and their properties. Focus on understanding the central limit theorem and confidence intervals — these are frequently tested. Always identify whether the random variable is discrete or continuous before choosing the approach.

Advanced Applications

Modern exams feature questions combining multiple quantitative concepts. Master these by practicing previous year papers.

Shortcut Methods

Percentages: Use fraction table (12.5%=1/8, 20%=1/5, 25%=1/4). SI/CI: For 2 years, CI-SI = P(r/100)^2. Time Work: Use LCM of time periods. Average Speed: 2ab/(a+b) for equal distances.

Pro Tip
Write key formulas on rough sheet immediately after the exam bell while memory is fresh.

Advanced Applications

Modern exams feature questions combining multiple quantitative concepts. Master these by practicing previous year papers.

Shortcut Methods

Percentages: Use fraction table (12.5%=1/8, 20%=1/5, 25%=1/4). SI/CI: For 2 years, CI-SI = P(r/100)^2. Time Work: Use LCM of time periods. Average Speed: 2ab/(a+b) for equal distances.

Pro Tip
Write key formulas on rough sheet immediately after the exam bell while memory is fresh.

Advanced Applications

Modern exams feature questions combining multiple quantitative concepts. Master these by practicing previous year papers.

Shortcut Methods

Percentages: Use fraction table (12.5%=1/8, 20%=1/5, 25%=1/4). SI/CI: For 2 years, CI-SI = P(r/100)^2. Time Work: Use LCM of time periods. Average Speed: 2ab/(a+b) for equal distances.

Pro Tip
Write key formulas on rough sheet immediately after the exam bell while memory is fresh.

Advanced Applications

Modern exams feature questions combining multiple quantitative concepts. Master these by practicing previous year papers.

Shortcut Methods

Percentages: Use fraction table (12.5%=1/8, 20%=1/5, 25%=1/4). SI/CI: For 2 years, CI-SI = P(r/100)^2. Time Work: Use LCM of time periods. Average Speed: 2ab/(a+b) for equal distances.

Pro Tip
Write key formulas on rough sheet immediately after the exam bell while memory is fresh.

Practice Questions

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