EXPECTED EVENT

Tags: #◇Computer Science #◇STT101A #◇Conceptual #◇Expected Event

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◇Overview

Mathematical Expectation (Expected Value), denoted $E[X]$ or $\mu$, is the probability-weighted average of all possible values of a random variable $X$. Rather than representing the single "most likely" outcome on one trial, $E[X]$ represents the long-run average value across infinitely many independent repetitions (by the Law of Large Numbers) or the center of mass of the probability distribution.

◇Key Definitions

• Discrete Expected Value: $E[X] = \sum_{x \in R_X} x \, P(X = x)$.
• Expected Value of a Function (LOTUS): $E[g(X)] = \sum_{x \in R_X} g(x) \, P(X = x)$.
• Indicator Random Variable ($I_A$): For an event $A$, let $I_A = 1$ if $A$ occurs and $0$ otherwise. Then $E[I_A] = 1 \cdot P(A) + 0 \cdot P(A^c) = P(A)$.
• Linearity of Expectation: $E[aX + bY + c] = aE[X] + bE[Y] + c$ for any random variables $X, Y$ (even if dependent!).

◇Worked Example

Problem: A cybersecurity bug bounty program pays $500 if a critical vulnerability is found (probability $0.05$), \char"24 100 for a minor bug (probability $0.20$), and \char"24 0 otherwise (probability $0.75$). If auditing takes $15 in cloud compute costs per run, what is the expected net profit per audit run?

Solution:
$E[\text{Payout}] = 500(0.05) + 100(0.20) + 0(0.75) = 25 + 20 + 0 = \$45$
$E[\text{Net Profit}] = E[\text{Payout} - 15] = 45 - 15 = \$30$

◇References & Navigation

• Previous Topic: ◇Basic Formula of Probability
• Next Topic: ◇Relative Frequency Form
• Related Notes: ◇Priori Probability, ◇Combinations