Notes › EENG 3421: Probability and Statistics for Engineers Lecture 11
Functions of a Random Variable
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Table of Contents
Derived Random Variables
- Sometimes we define a new random variable in terms of an existing one
- The new random variable is called a derived random variable
- The mapping is a deterministic function applied to the original random variable
Definition 11.1 (Derived Random Variable).
Each sample value $y$ of a derived random variable $Y$ is a mathematical function $g(x)$ of a sample value $x$ of another random variable $X$ We write
$$ Y = g(X) $$- The range of $Y$ is
- all possible values $g(x)$ where $x \in S_X$
- multiple values of $x$ may map to the same $y$
- When multiple $x$ map to the same $y$, the probabilities add
Theorem 11.2 (PMF of a Derived Random Variable).
For a discrete random variable $X$, the PMF of $Y=g(X)$ is
$$ P_Y(y) = \sum_{x \,:\, g(x)=y} P_X(x) $$Corollary 11.3 (One to One Case).
If $g$ is one to one on $S_X$, then for $y=g(x)$
$$ P_Y(y) = P_X(x) $$Corollary 11.4 (Many to One Case).
If $g(x_1)=g(x_2)=y$ with $x_1\ne x_2$, then
$$ P_Y(y)=P_X(x_1)+P_X(x_2) $$and similarly for more than two preimages
References
- Course lecture slides 16–17
Sources
- Course lecture slides 16–17