4. Probability Distributions: Random Variables
Expected Value of a Random Variable
- The expected value
- The variance or standard deviation
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:
\[\begin{array}{rcl}
\mathbb{E}[X]&=&\sum\limits_{\text{all }x\text{ in }R(X)}x\cdot f(x)\\
&=&1\cdot \mathbb{P}(X=1)+2\cdot \mathbb{P}(X=2) +\ldots+ 6 \cdot \mathbb{P}(X=6)\\
&=&1\cdot \cfrac{1}{6}+2\cdot \cfrac{1}{6}+3\cdot \cfrac{1}{6}+4\cdot \cfrac{1}{6}+5\cdot \cfrac{1}{6}+6\cdot \cfrac{1}{6}\\
&=& 3.5\\
\end{array}\]
#\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.