Question

Determine the mean and variance of the random variable with the following probability mass function.

Answer 1

Answer:

a. P(X = 2) = 0.2

b. P(X = 3) = 0.3

c. P(X > 2.5) = 0.7

d. P(X = 1) = 0.1

e. Mean = 3

f. Variance = 1

Step-by-step explanation:

As given,

Probability mass function (pmf) = $(\frac{1}{2})(\frac{x}{5} ) = \frac{x}{10}$

Now,

a. P(X = 2) = $\frac{2}{10}$ = 0.2

b. P(X = 3) = $\frac{3}{10}$ = 0.3

c. P(X > 2.5) = P(X = 3) + P(X = 4) = $\frac{3}{10} + \frac{4}{10}$ = 0.3 + 0.4 = 0.7

d. P(X = 1) = $\frac{1}{10}$ = 0.1

e. Mean = E(X) = $1.\frac{1}{10} + 2.\frac{2}{10} + 3.\frac{3}{10} + 4.\frac{4}{10}$ = 0.1 + 0.4 + 0.9 + 1.6 = 3

f. Variance = E(X²) - [ E(X) ]² = $1^{2} .\frac{1}{10} + 2^{2} .\frac{2}{10} + 3^{2} .\frac{3}{10} + 4^{2} .\frac{4}{10}$  - [3]²

                                             = 0.1 + 0.8 + 2.7 + 6.4 - 9

                                             = 10 - 9 = 1

∴ we get

a. P(X = 2) = 0.2

b. P(X = 3) = 0.3

c. P(X > 2.5) = 0.7

d. P(X = 1) = 0.1

e. Mean = 3

f. Variance = 1