Chapter 4. Probability Distributions: Random Variables
Sums of Random Variables
Sums of Random Variables
Suppose two independent random variables #Y# and #Z# are added to create a new variable #X#:
\[X = Y + Z\] Then the expected value and variance of #X# are:
\[\begin{array}{rcl}
\mathbb{E}[X] &=& \mathbb{E}[Y] + \mathbb{E}[Z] \\\\
\mathbb{V}[X] &=& \mathbb{V}[Y] + \mathbb{V}[Z]
\end{array}\]
Suppose two independent random variables #Y# and #Z# are subtracted to create a new variable #X#:
\[X = Y - Z\] Then the expected value and variance of #X# are:
\[\begin{array}{rcl}
\mathbb{E}[X] &=& \mathbb{E}[Y] - \mathbb{E}[Z]\\\\
\mathbb{V}[X] &=& \mathbb{V}[Y] + \mathbb{V}[Z]
\end{array}\]
Suppose a random variable #Y# is multiplied with a constant #k# to create a new variable #X#:
\[X = k \cdot Y\] Then the expected value, variance, and standard deviation of #X# are:
\[\begin{array}{rcl}
\mathbb{E}[X] &=& k \cdot \mathbb{E}[Y]\\\\
\mathbb{V}[X] &=& k^2 \cdot \mathbb{V}[Y] \\\\
\mathbb{SD}[X] &=& |k| \cdot \mathbb{SD}[Y]
\end{array}\]
Suppose:
#\,\,\,\,\,\,\,\,\scriptsize{\bullet}# #\,\mathbb{E}[Y]= 6.9\phantom{0}##\,\,\,\,\,\,\,\,\scriptsize{\bullet}# #\,\mathbb{V}[Y]= 1.88##\,\,\,\,\,\,\,\,\scriptsize{\bullet}# #\,\mathbb{E}[Z]= 6.0\phantom{0}##\,\,\,\,\,\,\,\,\scriptsize{\bullet}# #\,\mathbb{V}[Z]= 2.10#
#\,\,\,\,\,\,\,\,\scriptsize{\bullet}# #\,X= Y + 2Z#
Calculate #\mathbb{E}[X]#.
To calculate #\mathbb{E}[X]#, make use of the following properties:
- If #X = Y+Z#, then #\mathbb{E}[X] = \mathbb{E}[Y] + \mathbb{E}[Z]#
- If #X = k\cdot Y#, then #\mathbb{E}[X] = k \cdot \mathbb{E}[Y]#
Applying these properties to #X= Y + 2Z #, we get:
\[\begin{array}{rcl}
\mathbb{E}[X] &=& \mathbb{E}[Y + 2Z]\\\\
&=& \mathbb{E}[Y] + \mathbb{E}[2Z]\\\\
&=& 1\cdot \mathbb{E}[Y] + 2 \cdot \mathbb{E}[Z]\\\\
&=& 1\cdot 6.9 + 2 \cdot6.0\\\\
&=& 18.9
\end{array}\]
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