If a discrete random variable X takes values 0,1,2,3, . . with probability P ( X =x)= k (x+1) 5^ -x , where k is a constant, then P ( X =0) is

Step-by-step solution

Given that P ( X =x)= k (x+1) 5^ -x , where X =0,1,2,3, ...(i) aligned & Since, _ x=0 ^ P ( X =x)=1 & k _ x=0 ^ (x+1) 5^ -x =1 & k [1+2(5)⁻¹+3(5)⁻²+4(5)⁻³+ ]=1 & k [1+2 ( 1 5 )+3 ( 1 5 )^2+4 ( 1 5 )^3+ ]=1 aligned gathered k [ 1 1- 1 5 + 1 1 5 (1- 1 5 )^2 ]=1 k 25 16 =1 k = 16 25 P ( X =0)= 16 25 (0+1) ( 1 5 )^ ...[From (i)] = 16 25 gathered