Expected Value of a Random Variable

4. Probability Distributions: Random Variables

Expected Value of a Random Variable

The probability distribution of a random variable is summarized using two parameters:

The expected value
The variance or standard deviation

Expected Value

Definition

The expected value or mean of a random variable $X$ is the center of its probability distribution.

If you observe $X$ a very large number of times, the sample mean of all observations should be near the expected value.

Notation

$\mathbb{E}[X]$

Alternatives: $\mu$ or $\mu_X$

Expected Value of a Discrete Random Variable

Let $X$ be a discrete random variable with probability distribution $f(x)$ and range $R(X)$ .

Then the expected value of $X$ is calculated as follows:

$\mathbb{E}[X]=\sum_{\text{all }x\text{ in }R(X)}x\cdot f(x)$

Where $f(x)=\mathbb{P}(X=x)$ .

Roll a die once and let $X$ denote the number of upward-facing dots.

Calculate the expected value of $X$ .

The probability distribution of $X$ is:

$\begin{array}{c|cccccc} x&1&2&3&4&5&6\\ \hline \mathbb{P}(X = x)&\cfrac{1}{6}&\cfrac{1}{6}&\cfrac{1}{6}&\cfrac{1}{6}&\cfrac{1}{6}&\cfrac{1}{6}\\ \end{array}$
The expected value of $X$ is:

$\begin{array}{rcl} \mathbb{E}[X]&=&\sum\limits_{\text{all }x\text{ in }R(X)}x\cdot f(x)\\ &=&1\cdot \mathbb{P}(X=1)+2\cdot \mathbb{P}(X=2) +\ldots+ 6 \cdot \mathbb{P}(X=6)\\ &=&1\cdot \cfrac{1}{6}+2\cdot \cfrac{1}{6}+3\cdot \cfrac{1}{6}+4\cdot \cfrac{1}{6}+5\cdot \cfrac{1}{6}+6\cdot \cfrac{1}{6}\\ &=& 3.5\\ \end{array}$

New example

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Expected Value of a Continuous Random Variable

The expected value of a continuous random variable is computed using integral calculus and is beyond the scope of this course.