326 / 762
326
Chapter 8: Discrete Probability Distributions
V
(
X
) =
∑
}
= 350
2
1/5 + (-50)
2
4/5 – 30
2
= 25,600 (square dollars).
Example 8-8:
Find the variance for the profits in the two portfolios in
Example 8-5
.
Solution:
Let the profit for portfolio A be represented by the random
variable
X
, and let the profit for portfolio B be represented by the random
variable
Y
.
Then,
V
(
X
) = (-1,000)
2
0.2 + (-100)
2
0.1 + (300)
2
0.4 + (1,500)
2
0.2 +
(2,500)
2
0.1 – 460
2
= 1,100,400 (dollar square).
V
(
Y
) = (-2000)
2
0.2 + (-500)
2
0.1 + (1800)
2
0.3 + (2000)
2
0.3 +
(3500)
2
0.1 – 1040
2
= 3,140,400 (square dollars).
Now, if you select a portfolio based on the variance, then you should select
the one with the smaller variability, since this would involve lesser risk.
Thus, in this case, you should select portfolio A, since it has the smaller
variability.
We can also use the
Mean and Variance for a Discrete Distribution
workbook to help with the computations. For this example one will have to
do the computations separately for the
X
and
Y
variables.
Standard Deviation
It is easier to deal with a quantity that has the same units as the variable
itself. If we take the square root of a square unit, we will get the unit itself.
Thus, if we take the square root of the variance, called the standard
deviation, we will get a quantity that has the same unit as the random
variable.
Definition: Standard Deviation for a Discrete Random Variable
The standard deviation for a discrete random variable
X
is defined to be the
positive square root of the variance.