Question

Let X be a binomial random variable with parameters n and p. Show that E[1 / (X + 1)] = (1 - (1 - p)^(n + 1)) / ((n + 1) * p).

Answer 1

To find E[1 / (X + 1)], we can use the binomial probability mass function (PMF) and simplify the expression to (1 - (1 - p)^(n + 1)) / ((n + 1) * p).

Step-by-step explanation:

To find E[1 / (X + 1)], we first need to find the probability mass function (PMF) of X. The PMF of a binomial random variable X with parameters n and p is given by P(X = x) = C(n, x) * p^x * (1 - p)^(n - x), where C(n, x) is the binomial coefficient.

Next, we substitute P(X = x) into the expression E[1 / (X + 1)] = ∑ [1 / (x + 1) * P(X = x)]. Using the binomial PMF and simplifying the expression, we get E[1 / (X + 1)] = (1 - (1 - p)^(n + 1)) / ((n + 1) * p).