Chapter 4. Probability Distributions: Random Variables
Expected Value of a Random Variable
Every probability distribution has two important characteristics:
- The expected value
- The variance / standard deviation
Definition
The expected value or mean of a random variable #X# is the center of its probability distribution.
Notation
\[\mathbb{E}[X]\,\,\,\,\text{or}\,\,\,\,\mu\]
The expected value of a random variable #X# can be thought of as its long-term average, meaning that as we repeat an experiment over and over again the average value of #X# will approach the expected value.
The applet below allows you to simulate the random experiment of rolling two six-sided dice and counting the sum of the upward-facing dots.
The blue outline shows the probability distribution of the sum of two dice, the orange bars represent the frequency distribution of the sums of the simulated dice rolls.
As you increase the number of dice rolls, the shape of the frequency distribution will look more and more like the shape of the probability distribution, and the average sum of the simulated dice rolls will approach the expected value of #7#.
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: \[\begin{array}{rcl}
\mathbb{E}[X] &=&\displaystyle\sum_{\text{all }x\text{ in }R(X)}x\cdot f(x) \\\\
&=&\displaystyle\sum_{\text{all }x\text{ in }R(X)}x\cdot \mathbb{P}(X=x) \\\\
\end{array}\]
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:
\[\begin{array}{rcl}
\mathbb{E}[X]&=&\sum\limits_{\text{all }x\text{ in }R(X)}x\cdot f(x)\\\\
&=&1\cdot \mathbb{P}(X=1)+2\cdot \mathbb{P}(X=2) +\ldots+ 6 \cdot \mathbb{P}(X=6)\\\\
&=&1\cdot \cfrac{1}{6}+2\cdot \cfrac{1}{6}+3\cdot \cfrac{1}{6}+4\cdot \cfrac{1}{6}+5\cdot \cfrac{1}{6}+6\cdot \cfrac{1}{6}\\\\
&=& 3.5\\
\end{array}\]
#\phantom{0}#
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.
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