Notes › EENG 3421: Probability and Statistics for Engineers Lecture 12
Expected Value of a Derived Random Variable
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Table of Contents
- If $Y=g(X)$, we can compute $E[Y]$ without explicitly finding the PMF of $Y$
- This is useful when $g$ collapses many $x$ values into the same $y$ value
Theorem 12.1 (Expected Value of a Derived Random Variable).
Given a random variable $X$ with PMF $P_X(x)$ and the derived random variable $Y=g(X)$
$$ E[Y] = \mu_Y = \sum_{x \in S_X} g(x) P_X(x) $$Theorem 12.2 (Centered Random Variable Has Zero Mean).
For any random variable $X$
$$ E[X-\mu_X]=0 $$Proof. Let $g(X)=X-\mu_X$ and apply the derived expectation theorem
$$ E[g(X)] = \sum_{x\in S_X} (x-\mu_X)P_X(x) $$Split the sum
$$ \sum_{x\in S_X} xP_X(x) - \mu_X \sum_{x\in S_X} P_X(x) $$The first sum is $\mu_X$ by definition The second sum is $1$ So the result is $0$
Theorem 12.3 (Linearity for Affine Functions).
For any random variable $X$ and constants $a,b$
$$ E[aX+b]=aE[X]+b $$Corollary 12.4 (Shift Does Not Change Relative Centering).
$$
E[X+b]=E[X]+b
$$
Corollary 12.5 (Scaling Scales the Mean).
$$
E[aX]=aE[X]
$$
- In general
- $E[g(X)]$ depends on the full distribution of $X$
- it is not true that $E[g(X)]=g(E[X])$ unless $g$ is affine
Proposition 12.6 (Nonlinearity Warning).
In general
$$ E[g(X)] \ne g(E[X]) $$when $g$ is nonlinear
References
- Course lecture slides 16–17
Sources
- Course lecture slides 16–17