Find k For what value of k can the function fx 49 14k2kx s

For what value of k can the function f(x) = (4/9 - 1/4k^2)k^x serve as the value of the probability distribution of a random variable with the countable infinite range x = 0,1,2, ?

Solution

Summing f(x) from 0 to infinity, it is a geometric series with first term a1 = (4/9 - 1/4 k^2) and common ratio r = k.

Thus, normalizing,

Sum = a1 / (1-r) = 1

Sum = (4/9 - 1/4 k^2) / (1-k) = 1

4/9 - 1/4 k^2 = 1-k

16 - 9k^2 = 36 - 36k

9k^2 - 36k + 20 = 0

Thus,

(3k-2)(3k-20) = 0

Thus,

k = 2/3 or 20/3

But 20/3 cannot be as the sum will diverege.

Thus,

k = 2/3 or in decimal, 0.66666666667 [ANSWER]