Question

A discrete random variable \( X \sim B(n, p) \) (i.e., binomial distribution). Given: \[ P(X = 2) = P(X = 3) \] Then what is the mean of \( X \)?

• A. \( 2 - p \)
• B. \( 3 - p \)
• C. \( p - 2 \)
• D. \( p - 3 \)

Hint:

Use binomial probability definitions and simplify ratios when probabilities of two values are equal.

Answer 1

The Correct Option is B

We know for binomial distribution: \[ P(X = r) = {n \choose r} p^r (1 - p)^{n - r} \] Given: \[ P(X = 2) = P(X = 3) \Rightarrow {n \choose 2} p^2 (1 - p)^{n - 2} = {n \choose 3} p^3 (1 - p)^{n - 3} \] Cancel common terms: \[ \begin{align} \frac{n(n - 1)}{2} (1 - p) = \frac{n(n - 1)(n - 2)}{6} p \Rightarrow \frac{1 - p}{2} = \frac{(n - 2)}{6} p \Rightarrow 3(1 - p) = (n - 2)p \Rightarrow 3 - 3p = np - 2p \Rightarrow 3 = p(n + 1) \Rightarrow np = 3 - p \Rightarrow \text{Mean} = np = 3 - p \]