Random Variables

Tags: #◇Computer Science #◇STT101A #◇Conceptual #◇Random Variables

◇Overview

Random Variables are classified into Discrete and Continuous families based on the structure of their range (support) $R_X$ , and can be analyzed jointly using bivariate/multivariate distributions, covariance, and linear combinations.

◇Key Definitions

Feature	Discrete Random Variable	Continuous Random Variable
Support $R_X$	Finite or countably infinite ( $\{x_1, x_2, \dots\}$ )	Uncountable interval(s) of $\mathbb{R}$ (e.g., $[0, \infty)$ or $(-\infty, \infty)$ )
Distribution Function	PMF $p(x) = P(X = x)$	PDF $f(x) \ge 0$ with $P(a \le X \le b) = \int_a^b f(x)\,dx$
Point Probability $P(X = c)$	$p(c) \ge 0$ (can be strictly positive)	Always $0$ ( $\int_c^c f(x)\,dx = 0$ )
Expected Value $E[X]$	$\sum_{x \in R_X} x \, p(x)$	$\int_{-\infty}^{\infty} x \, f(x)\,dx$

Linear Scaling of Mean & Variance: For constants $a, b \in \mathbb{R}$ :
$E[aX + b] = aE[X] + b, \qquad \text{Var}(aX + b) = a^2 \text{Var}(X)$
Variance of a Sum: $\text{Var}(X \pm Y) = \text{Var}(X) + \text{Var}(Y) \pm 2\text{Cov}(X, Y)$ . If $X$ and $Y$ are independent, $\text{Cov}(X, Y) = 0$ , so $\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y)$ and $\text{Var}(X - Y) = \text{Var}(X) + \text{Var}(Y)$ .

◇Worked Example

Problem: Let $X$ and $Y$ be independent random variables with $E[X] = 10, \text{Var}(X) = 4$ and $E[Y] = 6, \text{Var}(Y) = 9$ . Find $E[2X - Y + 3]$ and $\text{Var}(2X - Y + 3)$ .

Solution:

$E[2X - Y + 3] = 2E[X] - E[Y] + 3 = 2(10) - 6 + 3 = 17$ .
$\text{Var}(2X - Y + 3) = 2^2 \text{Var}(X) + (-1)^2 \text{Var}(Y) + 0 = 4(4) + 1(9) = 16 + 9 = 25$ .

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Related Notes: ◇Bayes's Formula, ◇Probability Mass Function

◇Overview

◇Key Definitions

◇Worked Example

◇References & Navigation

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