Applied Statistics and Probability for Engineers

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

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.

The variance of a random variable Xcan be considered to be the expected value of a specific

function of X, namely,. In general, the expected value of any function

of a discrete random variable is defined in a similar manner.

h 1 X 2  1 X 22 h 1 X 2

 2 V 1 X 2  2 1.851.36

V 1 X 2  1021 0.08 2   1121 0.15 2  p  1521 0.07 2 12.5^2 1.85

E 1 X 2  101 0.08 2    111 0.15 2   p  151 0.07 2 12.5

f 1 x 2

If Xis a discrete random variable with probability mass function

E 3 h 1 X 24  a (3-4)

x

xh 1 x 2 f 1 x 2

f 1 x 2 ,

Expected Value of a

Function of a

Discrete Random

Variable

EXAMPLE 3-12 In Example 3-9, Xis the number of bits in error in the next four bits transmitted. What is the

expected value of the square of the number of bits in error? Now,. Therefore,

In the previous example, the expected value of does not equal squared. However, in

the special case that for any constants aand b, This

can be shown from the properties of sums in the definition in Equation 3-4.

EXERCISES FOR SECTION 3-4

h 1 X 2 aX b E 3 h 1 X 24 aE 1 X 2     b.

X^2 E 1 X 2

    32

 0.0036     42

 0.00010.52

E 3 h 1 X 24  02

 0.6561     12

 0.2916     22

 0.0486

h 1 X 2 X^2

3-37. If the range of Xis the set {0, 1, 2, 3, 4} and P(X

x) 0.2 determine the mean and variance of the random variable.

3-38. Determine the mean and variance of the random vari-

able in Exercise 3-13.

3-39. Determine the mean and variance of the random vari-

able in Exercise 3-15.

3-40. Determine the mean and variance of the random vari-

able in Exercise 3-17.

3-41. Determine the mean and variance of the random vari-

able in Exercise 3-19.

3-42. Determine the mean and variance of the random vari-

able in Exercise 3-20.

3-43. Determine the mean and variance of the random vari-

able in Exercise 3-22.

3-44. Determine the mean and variance of the random vari-

able in Exercise 3-23.

3-45. The range of the random variable Xis

where xis unknown. If each value is equally likely and the

mean of Xis 6, determine x.

3 0, 1, 2, 3, x 4 ,

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