Show that the following function is a probability mass funct

Show that the following function is a probability mass function (pmf) for the random variable X and constant b. f(x) = (1/1 + b)(b/1 + b)^x, x = 0, 1, 2, 3, . . .

Solution

Here, f(x) must satisfy

f(0) + f(1) + f(2) + f(3) ... = 1

However, we see that the series on the left is a geomtric series with first term a1 = 1/(1+b) and common ratio r = b/(1+b).

As the sum of a geometric series is

S = a1 / (1-r), then

f(0) + f(1) + f(2) + f(3) ... = [1/(1+b)]/[1 - b/(1+b)]

f(0) + f(1) + f(2) + f(3) ... = [1/(1+b)]/[(1+b-b)/(1+b)]

f(0) + f(1) + f(2) + f(3) ... = [1/(1+b)]/[1/(1+b)]

f(0) + f(1) + f(2) + f(3) ... = 1 [SATISFIED!]