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Random Variables

Short Notes

A random variable is a function that assigns a numerical value to each outcome in a sample space.

  • Discrete Random Variable: Takes countable values (e.g., 0, 1, 2, ...)
  • Continuous Random Variable: Takes any value in an interval

Key Theories & Formulas

1. Probability Mass Function (PMF) - Discrete

\[P(X = x) = p(x)\]

Properties:

  • \(p(x) \geq 0\) for all x
  • \(\sum_{x} p(x) = 1\)

2. Probability Density Function (PDF) - Continuous

\[f(x) \geq 0\]

Properties:

  • \(\int_{-\infty}^{\infty} f(x) \, dx = 1\)
  • \(P(a \leq X \leq b) = \int_a^b f(x) \, dx\)
  • \(P(X = a) = 0\) for continuous variables

3. Cumulative Distribution Function (CDF)

\[F(x) = P(X \leq x) = \begin{cases} \sum_{t \leq x} p(t) & \text{discrete} \\ \int_{-\infty}^x f(t) \, dt & \text{continuous} \end{cases}\]

Properties:

  • \(0 \leq F(x) \leq 1\)
  • F is non-decreasing
  • \(\lim_{x \to -\infty} F(x) = 0\), \(\lim_{x \to \infty} F(x) = 1\)
  • \(P(a < X \leq b) = F(b) - F(a)\)

4. Expected Value (Mean)

\[E[X] = \mu = \begin{cases} \sum_x x \cdot p(x) & \text{discrete} \\ \int_{-\infty}^{\infty} x \cdot f(x) \, dx & \text{continuous} \end{cases}\]

Properties:

  • \(E[c] = c\) (constant)
  • \(E[aX + b] = aE[X] + b\)
  • \(E[X + Y] = E[X] + E[Y]\) (always true)
  • \(E[XY] = E[X] \cdot E[Y]\) (only if X, Y independent)

5. Variance and Standard Deviation

\[Var(X) = \sigma^2 = E[(X - \mu)^2] = E[X^2] - (E[X])^2\]

Properties:

  • \(Var(c) = 0\)
  • \(Var(aX + b) = a^2 \cdot Var(X)\)
  • \(Var(X + Y) = Var(X) + Var(Y)\) (if X, Y independent)

$\(\sigma = \sqrt{Var(X)}\)$ (Standard Deviation)

6. Higher Moments

  • n-th moment about origin: \(E[X^n]\)
  • n-th moment about mean: \(E[(X - \mu)^n]\)
  • Coefficient of Variation: \(CV = \frac{\sigma}{\mu}\)

7. Moment Generating Function (MGF)

\[M_X(t) = E[e^{tX}]\]

Useful property: \(E[X^n] = M_X^{(n)}(0)\) (n-th derivative at t=0)

Example Problems

Problem 1: A random variable X has PMF: P(X=1) = 0.2, P(X=2) = 0.5, P(X=3) = 0.3. Find E[X] and Var(X).

Solution:

  1. \(E[X] = 1(0.2) + 2(0.5) + 3(0.3) = 0.2 + 1 + 0.9 = \mathbf{2.1}\)
  2. \(E[X^2] = 1(0.2) + 4(0.5) + 9(0.3) = 0.2 + 2 + 2.7 = 4.9\)
  3. \(Var(X) = E[X^2] - (E[X])^2 = 4.9 - 4.41 = \mathbf{0.49}\)

Problem 2: If X has PDF \(f(x) = 2x\) for \(0 \leq x \leq 1\), find E[X].

Solution: $\(E[X] = \int_0^1 x \cdot 2x \, dx = 2\int_0^1 x^2 \, dx = 2 \cdot \frac{x^3}{3}\Big|_0^1 = \frac{2}{3}\)$

Problem 3: If E[X] = 5 and Var(X) = 4, find E[(2X + 3)²].

Solution:

  1. \(E[(2X + 3)^2] = E[4X^2 + 12X + 9]\)
  2. \(= 4E[X^2] + 12E[X] + 9\)
  3. \(E[X^2] = Var(X) + (E[X])^2 = 4 + 25 = 29\)
  4. \(= 4(29) + 12(5) + 9 = 116 + 60 + 9 = \mathbf{185}\)

Hardest GATE Questions

Topic: CDF to PDF Conversion

Question (GATE 2017 Style): A random variable X has CDF: $\(F(x) = \begin{cases} 0 & x < 0 \\ x^2 & 0 \leq x \leq 1 \\ 1 & x > 1 \end{cases}\)$ Find P(0.5 < X < 0.8).

Solution:

  1. \(P(0.5 < X < 0.8) = F(0.8) - F(0.5)\)
  2. \(= (0.8)^2 - (0.5)^2 = 0.64 - 0.25 = \mathbf{0.39}\)

Topic: Independence and Expectation

Question (GATE 2018 Variant): X and Y are independent random variables with E[X] = 2, E[Y] = 3, Var(X) = 1, Var(Y) = 4. Find E[(X-Y)²].

Solution:

  1. \(E[(X-Y)^2] = Var(X-Y) + (E[X-Y])^2\)
  2. \(Var(X-Y) = Var(X) + Var(Y) = 1 + 4 = 5\) (independent)
  3. \(E[X-Y] = E[X] - E[Y] = 2 - 3 = -1\)
  4. \(E[(X-Y)^2] = 5 + 1 = \mathbf{6}\)

Topic: MGF Application

Question: If the MGF of X is \(M_X(t) = e^{3t + 2t^2}\), find E[X] and Var(X).

Solution:

  1. \(M_X'(t) = (3 + 4t)e^{3t + 2t^2}\)
  2. \(E[X] = M_X'(0) = 3 \cdot 1 = \mathbf{3}\)
  3. \(M_X''(t) = 4e^{3t+2t^2} + (3+4t)^2 e^{3t+2t^2}\)
  4. \(E[X^2] = M_X''(0) = 4 + 9 = 13\)
  5. \(Var(X) = 13 - 9 = \mathbf{4}\)

References