Sums of Random Variables

4. Probability Distributions: Random Variables

Sums of Random Variables

Sum of Independent Random Variables

Suppose two independent random variables #Y# and #Z# are added to give a new variable #X#. Then their sum is also a random variable, with the following expected value and variance:

#\mathbb{E}[X] = \mathbb{E}[Y] + \mathbb{E}[Z]#
#Var[X] = Var[Y] + Var[Z]#

If, in contrast, #Y# and #Z# are subtracted, then the expected value and variance new variable #X# are

#\mathbb{E}[X] = \mathbb{E}[Y] - \mathbb{E}[Z]#
#Var[X] = Var[Y] + Var[Z]#

If a random variable #Y# is multiplied with a constant #p#, the resulting variable #X# has the following properties.

#\mathbb{E}[X] = p \cdot \mathbb{E}[Y]#
#Var[X] = p^2 \cdot Var[Y]#
#SD[X] = |p| \cdot SD[Y]#

Suppose:

#\,\,\,\,\,\,\,\,\scriptsize{\bullet}# #\,\mathbb{E}[Y]= 5.3\phantom{0}##\,\,\,\,\,\,\,\,\scriptsize{\bullet}# #\,Var[Y]= 1.66#

#\,\,\,\,\,\,\,\,\scriptsize{\bullet}# #\,\mathbb{E}[Z]= 5.1\phantom{0}##\,\,\,\,\,\,\,\,\scriptsize{\bullet}# #\,Var[Z]= 1.21#

#\,\,\,\,\,\,\,\,\scriptsize{\bullet}# #\,X= 2Y + 5Z#

Find #\mathbb{E}[X]#.

#\mathbb{E}[X]=36.1#

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= 2Y + 5Z #, we get:
\[\begin{array}{rcl}
\mathbb{E}[X] &=& \mathbb{E}[2Y + 5Z]\\
&=& \mathbb{E}[2Y] + \mathbb{E}[5Z]\\
&=& 2\cdot \mathbb{E}[Y] + 5 \cdot \mathbb{E}[Z]\\
&=& 2\cdot 5.3 + 5 \cdot5.1\\
&=& 36.1
\end{array}\]

New example

Sum of Dependent Random Variables

Suppose two dependent random variables #Y# and #Z# are added to give a new variable #X#. Then their sum is also a random variable, with the following expected value and variance:

#\mathbb{E}[X] = \mathbb{E}[Y] + \mathbb{E}[Z]#
#Var[X] = Var[Y] + Var[Z] + 2Cov[Y,Z]#

If, in contrast, #Y# and #Z# are subtracted, then the expected value and variance new variable #X# are

#\mathbb{E}[X] = \mathbb{E}[Y] - \mathbb{E}[Z]#
#Var[X] = Var[Y] + Var[Z] - 2Cov[Y,Z]#

The above behaviour follows form the general rule for adding variablels #Y# and #Z# multiplied by respectively constants #p# and #q# to give a new variable #X#:

#\mathbb{E}[X] = p \cdot \mathbb{E}[Y] + q \cdot \mathbb{E}[Z]#
#Var[X] = p^2 \cdot Var[Y] + q^2 \cdot Var[Z] + 2pq \cdot Cov[Y,Z]#