A random variable X takes the values. 0,1,2,3,........ with probability P ( X =x)= k (x+1) ( 1 5 )^x , where k is a constant. Then P ( X =0) is

Step-by-step solution

To determine P(X=0) , the normalizing constant k must first be found using the probability distribution constraint. The sum over all x gives _ x=0 ^ k(x+1) ( 1 5 )^x = 1 . The series _ x=0 ^ (x+1)r^x converges to 1 (1-r)^2 for |r| This gives k 25 16 = 1 , so k = 16 25 . Then P(X=0) = k(0+1) ( 1 5 )^0 = k 1 1 = 16 25 . The value matches option A .