TL;DR
The probability-weighted average of all possible outcomes of a random variable: \( E[X] = \sum_x x \cdot P(X=x) \) for discrete variables, or \( E[X] = \int x f(x) dx \) for continuous variables.
Expected Value
The probability-weighted average of all possible outcomes of a random variable: \( E[X] = \sum_x x \cdot P(X=x) \) for discrete variables, or \( E[X] = \int x f(x) dx \) for continuous variables.
Why it matters for interviews
The most fundamental concept in quant interviews. Every pricing formula, strategy evaluation, and risk metric is built on expectations. Linearity of expectation is the single most powerful problem-solving tool.
Definition and Mathematical Foundation
The probability-weighted average of all possible outcomes of a random variable: \( E[X] = \sum_x x \cdot P(X=x) \) for discrete variables, or \( E[X] = \int x f(x) dx \) for continuous variables.
Expected value satisfies several key properties: linearity (\( E[aX+bY] = aE[X]+bE[Y] \) regardless of dependence), monotonicity (\( X \leq Y \) implies \( E[X] \leq E[Y] \)), and the law of the unconscious statistician (\( E[g(X)] = \int g(x)f(x)dx \)). For discrete variables, the sum may not converge absolutely, in which case the expectation is undefined.
Application in Quantitative Finance
The most fundamental concept in quant interviews. Every pricing formula, strategy evaluation, and risk metric is built on expectations. Linearity of expectation is the single most powerful problem-solving tool.
Related Terms
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