Page 91 - Applied Statistics And Probability For Engineers

PQ220 6234F.Ch 03 13/04/2002 03:19 PM Page 69

3-4 MEAN AND VARIANCE OF A DISCRETE RANDOM VARIABLE 69

EXAMPLE 3-11 The number of messages sent per hour over a computer network has the following distribution:

x number of messages 10 11 12 13 14 15
f 1x2 0.08 0.15 0.30 0.20 0.20 0.07

Determine the mean and standard deviation of the number of messages sent per hour.

E1X2 1010.082 1110.152 p 1510.072 12.5
2
2
2
2
V1X2 10 10.082 11 10.152 p 15 10.072 12.5 1.85
2V1X2 21.85 1.36
The variance of a random variable X can be considered to be the expected value of a specific
function of X, namely, h1X2 1X 2 2 . In general, the expected value of any function h1X2
of a discrete random variable is defined in a similar manner.

Expected Value of a
Function of a If X is a discrete random variable with probability mass function f 1x2,
Discrete Random
Variable
E3h1X24 a xh1x2 f 1x2 (3-4)
x

EXAMPLE 3-12 In Example 3-9, X is the number of bits in error in the next four bits transmitted. What is the
2
expected value of the square of the number of bits in error? Now, h1X2 X . Therefore,
2
2
2
E3h1X24 0
0.6561 1
0.2916 2
0.0486
2
2
3
0.0036 4
0.0001 0.52
2
In the previous example, the expected value of X does not equal E1X2 squared. However, in
the special case that h1X2 aX b for any constants a and b, E3h1X24 aE1X2 b. This
can be shown from the properties of sums in the definition in Equation 3-4.

EXERCISES FOR SECTION 3-4

3-37. If the range of X is the set {0, 1, 2, 3, 4} and P(X 3-42. Determine the mean and variance of the random vari-
x) 0.2 determine the mean and variance of the random variable. able in Exercise 3-20.
3-38. Determine the mean and variance of the random vari- 3-43. Determine the mean and variance of the random vari-
able in Exercise 3-13. able in Exercise 3-22.
3-39. Determine the mean and variance of the random vari- 3-44. Determine the mean and variance of the random vari-
able in Exercise 3-15. able in Exercise 3-23.
3-40. Determine the mean and variance of the random vari- 3-45. The range of the random variable X is 30, 1, 2, 3, x4,
able in Exercise 3-17. where x is unknown. If each value is equally likely and the
3-41. Determine the mean and variance of the random vari- mean of X is 6, determine x.
able in Exercise 3-19.