3.3.3. Mean and variance of linear function of a random variable
Let X be a random variable that takes the value x with probability P(x) and consider a new random variable Y, defined by Y=a+bX.
Suppose
that random variable X
has mean
,
and variance
.
Then mean and variance of Y are
and
so that standard deviation of
Y
is
.
Example:
A car salesman estimates the following probabilities for the number of cars that he will sell in next month.
Number cars |
0 1 2 3 4 |
Probability |
0.12 0.20 0.25 0.25 0.18 |
a) Find the expected number of cars that will be sold in the next month.
b) Find the standard deviation of the number of cars that will be sold in
next month.
c) The salesperson receives for the month a salary of $300, plus an additional $200 for each car sold. Find the mean and standard deviation of his total monthly salary.
Solution:
a) The random variable X has mean
.
b)Variance
1.621
.
c) Total
monthly salary of salesperson can be written as
.
Then
$734.
.
=$254.64.
Summary results for the mean and variance of special linear functions:
a)
Let b=0
in the linear function,
.
Then
for any constant a.
and Var (a) =0
If a random variable always takes the value a, it will have a mean a and
a variance 0.
b)
Let
in the linear function,
.
Then
.
and
.
Exercises
1. Find the mean and standard deviation for each of the following probability distributions.
X |
P(x) |
0 1 2 3 |
0.12 0.27 0.43 0.18 |
X |
P(x) |
6 7 8 9 |
0.36 0.26 0.21 0.17 |
2. Given the following probability distribution.
Find
X |
P(x) |
0 1 2 3 |
0.4 0.3 0.2 0.1 |
3.
Let x be the number of errors that a randomly selected page of a book contains. The following table lists the probability distribution of x.
-
X
0 1 2 3 4
P(x)
0.73 0.16 0.06 0.04 0.01
Find the mean and standard deviation.
4.
Suppose the probability function of a random variable X
is given by the formula
for x
=2, 3, 4, 5
Calculate the mean and standard deviation of this distribution.
5. Given the two probability distributions
X |
P(x) |
1 2 3 |
0.2 0.6 0.2 |
X |
P(x) |
0 1 2 3 4 |
0.1 0.2 0.4 0.2 0.1 |
a) Verify that both distributions have the same mean.
b) Compare the two standard deviations.
6. An instant lottery ticket costs $2. Out of a total of 10 000 tickets printed for this lottery, 1000 tickets contain a prize of $5 each, 100 tickets have a prize of $10 each, 5 tickets have a prize of $1000 each, and 1 ticket has a prize of $5000. Let x be the random variable that denotes the net amount a player wins by playing this lottery. Write the probability distribution of x. Determine the mean and standard deviation of x. How will you interpret the values of the mean and standard deviation of x?
7. A TV repairer estimates the probabilities for the number of hours required to complete some job as follows:
-
Time taken (Hours)
1 2 3 4 5
Probability
0.05 0.2 0.35 0.3 0.1
a) Find the expected time to complete the job.
b) The TV repairer’s service is made up of two parts- a fixed cost of $20, plus $2 for each hour taken to complete the job. Find the mean and standard deviation of total cost.
8. Consider the following probability distribution for the random variable X.
X |
P(x) |
10 20 30 40 |
0.20 0.40 0.25 0.15 |
a) Find the expected value of X.
b) Find the variance and standard deviation.
c) If
,
find the expected value, variance, and standard deviation for Y.
Answers
1. a) 1.67; 0.906; b) 7.19; 1.102; 2. 1; 1; 1; 3. 0.44; 0.852; 4. 3.12; 1.09;
5.
a)
;
b)
;
;6.
1.6; 54.78; 7.
a) 3.2;
b) 26.4; 2.06; 8. a) 23.5; b) 92.75 and 9.63; c) 75.5; 834.75; 28.89.