If the mean and the variance of a binomial variate X are 1 and 0.75 respectively, then which of the following is true?

Step-by-step solution

Given mean np = 1 and variance npq = 0.75 = 3 4 . Dividing variance by mean: q = npq np = 3 4 Since p + q = 1 , p = 1 - 3 4 = 1 4 . Substituting p in np = 1 : n ( 1 4 ) = 1 n = 4 The probability of X = r is given by P(X = r) = ^ n C_ r p^r q^ n-r . Calculating P(X = 1) and P(X = 2) : P(X = 1) = ⁴C₁ ( 1 4 )^1 ( 3 4 )^3 = 4 1 4 27 64 = 27 64 P(X = 2) = ⁴C₂ ( 1 4 )^2 ( 3 4 )^2 = 6 1 16 9 16 = 54 256 = 27 128 Comparing the two probabilities: 2P(X = 2) = 2 27 128 = 27 64 = P(X = 1) Thus, P(X = 1) = 2P(X = 2) . Answer: P