Question

A random variable X can take all non-negative integral values and the probability that X takes the value r is proportional to 5^−𝑟. Find P (X < 3).

Answer 1

Given `P(X = r)alpha1/(5^r)`

`P(X = r) = k1/(5^r)`, (where k is a non-zero constant)

`P(r = 0) = k*1/5^0`

`P(r = 1) = k*1/5^1`

`P(r = 2) = k*1/5^2`

`P(r = 3) = k*1/5^3`

We have, P(X = 0) + P(X = 1) + P(X = 2) + ....... = 1

⇒ `k (1 + 1/5 + 1/5^2 + 1/5^3 + .......) = 1`

⇒ `k (1/(1 - 1/5)) = 1`

⇒ `k = 4/5`

So, P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)

= `4/5 (1 + 1/5 + 1/5^2)`

= `4/5 ((25 + 5 + 1)/25)`

 = `124/125`