Question

Complete the following two problems

(a) Show that P(X = x) =(1/2)^ x+1 for x = 0,1,2,... is a valid probability mass function for the discrete random variable X.

(b) Given the following cumulative distribution function, nd the probability mass function.

x

P(X ≤ x)

0	0.05
1	0.32
2	0.64
3	0.95
4	1

Answer 1

a) To be a valid probability distribution, the sum of all probability must be equal to one.

P(X = x) = 0.5^x+1

sum = 0.5¹ + 0.5² + 0.5³ + ....

        = 0.5 / (1 - 0.5)                 (sum of infinte GP formula)

        = 1

Since the sum is equal to one, this is a valid probaility distribution.

b) P(X = 0) = 0.05

P(X = 1) = 0.32 - 0.05 = 0.27

P(X = 2) = 0.64 - 0.32 = 0.32

P(X = 3) = 0.95 - 0.64 = 0.31

P(X = 4) = 1 - 0.95 = 0.05