Random Variables, Expectation, Variance & Covariance
“PDF/CDF properties, Linearity of Expectation, Covariance, and Chebyshev's inequality.”
1.Random Variables, Moments, and Joint Distributions
§ 4.1.A random variable X is a measurable function mapping the sample space S to the real line R.
Discrete vs Continuous:
- Discrete: Takes values in a countable set. Characterized by Probability Mass Function (PMF) p(x) = P(X = x).
- Continuous: P(X = c) = 0 for any individual point c. Probability is assigned over intervals via Probability Density Function (PDF) f(x).
Linearity of Expectation Theorem:
For any random variables X_1, X_2, ..., X_n and constants c_1, c_2, ..., c_n:
E[ Sum_{i=1}^n c_i X_i ] = Sum_{i=1}^n c_i E[X_i]
This requires NO assumption of independence. This is one of the most powerful problem-solving tools in computer science algorithm analysis and GATE.
To find the expected number of events that occur, define indicator I_i = 1 if event i occurs, 0 otherwise. Then E[I_i] = P(event i occurs). By Linearity of Expectation, E[Total] = Sum E[I_i] = Sum P(event i occurs). This solves hat-check problems, coupon collectors, and hash collision expectations in 2 lines.
2. Worked Problem: Finding PDF Normalization Constant and Variance
Remember to integrate over the valid support interval only, and do not forget to square E[X] when subtracting from E[X^2].
Problem [Original Practice Problem · Modeled on GATE Pattern]:
A continuous random variable X has probability density function:
f(x) = c * (2x - x^2) for 0 <= x <= 2, and 0 otherwise.
Find (a) the constant c, (b) E[X], and (c) Var(X).
Step 1: Determine c using the total probability axiom integral f(x) dx = 1:
integral_0^2 c * (2x - x^2) dx = 1
c * [ x^2 - x^3 / 3 ]_0^2 = 1
c [ 4 - 8/3 ] = c [ 4/3 ] = 1 => c = 3/4.
Step 2: Compute Expectation E[X]:
E[X] = integral_0^2 x f(x) dx = (3/4) integral_0^2 (2x^2 - x^3) dx
E[X] = (3/4) * [ (2x^3 / 3) - (x^4 / 4) ]_0^2
E[X] = (3/4) [ (16/3) - (16/4) ] = (3/4) [ 16/3 - 4 ] = (3/4) * [ 4/3 ] = 1.
(Notice f(x) is symmetric around x = 1, so E[X] = 1 by symmetry!).
Step 3: Compute E[X^2]:
E[X^2] = integral_0^2 x^2 f(x) dx = (3/4) integral_0^2 (2x^3 - x^4) dx
E[X^2] = (3/4) * [ (2x^4 / 4) - (x^5 / 5) ]_0^2
E[X^2] = (3/4) [ (32/4) - (32/5) ] = (3/4) [ 8 - 6.4 ] = (3/4) * (1.6) = 1.2 = 6/5.
Step 4: Compute Variance Var(X):
Var(X) = E[X^2] - (E[X])^2 = (6/5) - (1)^2 = 1/5 = 0.20.
Primary Academic References & Exam Sources
Every concept in Notes From a B.Tech Brain is synthesized from authoritative academic texts, seminal peer-reviewed research, verified exam archives, and renowned university lectures.
Curated for clarity, rigorous numerical solving, and semester excellence:
Efficient GATE maths coverage with high-yield topic prioritization.
Building geometric intuition for linear algebra before solving problems.