A= X B(3,1/6)
B= X Geo (1/2)
Find
i) P(B>2)
ii) P(A=B)
thanks
Do you mean that A is a random variable that has a binomial distribution with n = 3 and p = 1/6?
Do oyu mean that B is a random variable that has a geometric distribution with p = 1/2?
If so, then:
i) Two options:
1. $\displaystyle \Pr(B > 2) = 1 - \Pr(B = 1) - \Pr(B = 2) = 1 - \frac{1}{2} - \frac{1}{4} = \frac{1}{4}$.
2. From the cdf, $\displaystyle \Pr(B > 2) = 1 - \Pr(B \leq 2) = 1 - \left[1 - \left(1 - \frac{1}{2}\right)^2\right] = \left(\frac{1}{2}\right)^2 = \frac{1}{4}$.
ii) Pr(A = B) = Pr(A = 1 | B = 1) Pr(B = 1) + Pr(A = 2 | B = 2) Pr(B = 2) + Pr(A = 3 | B = 3) Pr(B = 3)
Assuming A and B are independent:
= Pr(A = 1) Pr(B = 1) + Pr(A = 2) Pr(B = 2) + Pr(A = 3) Pr(B = 3)
$\displaystyle = \left[ 3 \left(\frac{1}{6}\right) \left(\frac{5}{6}\right)^2 \right] \left(\frac{1}{2}\right) + \left[ 3 \left(\frac{1}{6}\right)^2 \left(\frac{5}{6}\right) \right] \left(\frac{1}{2}\right)^2 + \left[\left(\frac{1}{6}\right)^3 \right] \left(\frac{1}{2}\right)^3 = .....$