Suppose the probability distribution px of random variable X

Suppose the probability distribution p(x) of random variable X is:

x

7

5

9

11

P(x)

.10

.30

.40

.20

a. Calculate the mean and standard deviation of X.
b. Let f(x) = (5x + 7). Calculate the probability distribution of f(x).
c. Calculate the mean and standard deviation of f(x). Use the \"short method.\"

Solution

Consider the table:

x   P(x)   x P(x)   x^2 P(x)
7   0.1   0.7   4.9
5   0.3   1.5   7.5
9   0.4   3.6   32.4
11   0.2   2.2   24.2

Sum [xP(x)] = 8
Sum[x^2 P(x)] = 69

As

Mean = Sum(x P(x))

Summing column 3,

Mean = 8 [ANSWER]

The standard deviation meanwhile is

standard deviation =sqrt{ Sum (x^2 P(x)) - [Sum (x P(x)]^2 }

= sqrt(69 - 8^2)

= 2.236067977 [ANSWER, standard deviation]

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b.

Substituting the values of f(x) = 5x + 7 to the x column,

f(x)   P[f(x)]
42   0.1
32   0.3
52   0.4
62   0.2

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c.

Mean (5x + 7) = 5 Mean (x) + 7 = 5(8) + 7 = 47 [ANSWER]

s (5x + 7) = s(5x) = 5 s(x) = 5*2.236067977 = 11.18033989 [ANSWER]