Question

Suppose that total 5 independent trials having a common probability of success 1/3 are performed. If X is the number of successes in the first2 trials, and Y is the number of successes in the final 3 trials, then X and Y are independent, since knowing the number of successes in the first 2 trials does not affect the distribution of the number of successes in the final 3 trials (by the assumption of independent trials). Find the joint p.d.f. of P(X = 2, Y = 1). ︵ 10 243 40 243 81 81

Answer 1

P(X = 2, Y = 1) = P(X = 2) times P(Y = 1) = binom(2 2) times (1/3)^2 times binom(3/1) times 1/3 times (1 - 1/3)^2 Rightarrow P(X = 2, Y = 1) = (1/3)^2 times 3 times 1/3 times (2/3)^2 Rightarrow P(X = 2, Y = 1) = 4/81

$\Rightarrow P(X=2,Y=1)= (\frac{1}{3})^2*3*\frac{1}{3}*(\frac{2}{3})^2$

$\Rightarrow P(X=2,Y=1)= \frac{4}{81}$ (ans)