MATH 180A Lecture 13

Theorem: Expectation is linear: If $X,Y$ are random variables defined on the same probability space and $c\in \mathbb{R}$, then

Proof.

Example. We return to . Calculate .

Functions of Discrete Random Variables

A biased coin lands heads 2/3 of the time. I flip it repeatedly until it lands heads. You win 2^k$ if it lands heads for the first time on the kth flip. How many dollars are you expected to win?

W = # of dollars we win

where N = # of coin flips Thus, where

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By definition, if is a discrete random variable and is a function, then is also a discrete random variable and

Theorem: If $X$ is a discrete random variable and $g$ is a function, then

Example. A biased coin lands heads 75% of the time. You flip it 4 times. If it lands heads an even number of times, I win 2. How many dollars am I expected to win?

W = # of dollars I win. where X = # of heads,

Continuous Random Variables

Expectation of discrete r.v. .

Expectation

Let be a continuous random variable, then its expectation is defined by is the pdf

Examples.

1. Let . Calculate .
2. A dart hits a circular board of radius 1 uniformly at random. What is the expected distance from the point it hits to the center of the board? distance to center