A random variable X has the following probability distribution: X -2 -1 0 1 2 P(X) k^2 2k k 2k 5k^2 The conditional probability P(X 0 X 1) is equal to

Step-by-step solution

The sum of all probabilities must be equal to 1 . k^2 + 2k + k + 2k + 5k^2 = 1 6k^2 + 5k - 1 = 0 (6k - 1)(k + 1) = 0 k = 1 6 or k = -1 . Since probabilities must be non-negative, k = -1 is rejected. Thus, k = 1 6 . We are asked to find the conditional probability P(X 0 X 1) . By definition, P(A B) = P(A B) P(B) . P(X 0 X 1) = P(0 X 1) P(X 1) First, calculate the numerator: P(0 X 1) = P(X=0) + P(X=1) = k + 2k = 3k Substituting k = 1 6 , we get 3 ( 1 6 ) = 1 2 . Next, calculate the denominator: P(X 1) = 1 - P(X=2) =