If the probability function of a random variable X is defined by P(X=k)=a ( k+1 2^k ) for k=0,1,2,3,4,5 , then the probability that X takes a prime value is

Step-by-step solution

We have, P(x=k)=a ( k+1 2^k ) We know that, gathered P(x=k)=1 P(x=0)+P(x=1) +P(x=2)+P(x=3)+P(x=4)+P(x=5)=1 gathered aligned & a+ 2 a 2 + 3 4 a+ 4 8 a+ 5 16 a+ 6 32 a=1 & a+a+ 3 4 a+ 1 2 a+ 5 16 a+ 3 16 a=1 & 16 a+16 a+12 a+8 a+5 a+3 a 16 =1 & 60 a=16 & a= 16 60 & a= 4 15 & Required probability & =P(x=2)+P(x=3)+P(x=4)= 3 a 4 + 1 a 2 + 3 a 16 & = 12 a+8 a+3 a 16 = 23 a 16 = 23 16 4 15 = 23 60 . & aligned