Let a random variable X have a binomial distribution with mean 8 and variance 4. If `P(X le 2 ) = (k)/(2^(16))`, then k is equal to

Options

A. 17
B. 121
C. 1
D. 137

Answer

D

Explanation

Let for the given random variable 'X' the binomial probability distribution have n-number of independent trials and probability of success and failure are p and q respectively. According to the question,
Mean =np =8 and variance=npq=4
`:. Q=(1)/(2)rArr p=1-q=(1)/(2)` Now, `nxx(1)/(2)=8rArr n=16`
`P(X=r)=^(16)C_(r )((1)/(2))^(16)`
`:. P(X le 2)=P(X=0)+P(X=1)+P(X=2)`
`=^(16)C_(0)((1)/(2))^(16)+overset(16)""C_(1)((1)/(2))^(16)+overset(16)""C_(2)((1)/(2))^(16)`
`=(1+16)+120)/(2^(16))=(137)/(2^(16))=(k)/(2^(16)) " "` (given)
`rArr " "k=137`