Question

Let \( (X_1, X_2, X_3) \) follow the multinomial distribution with the number of trials being 100 and the probability vector \( \left( \frac{3}{10}, \frac{1}{10}, \frac{3}{5} \right) \). 
Then \( E(X_2 | X_3 = 40) \) equals:

• A. 25
• B. 15
• C. 30
• D. 45

Hint:

In multinomial distributions, conditional expectations can often be treated as binomial distributions, where the total number of trials is fixed and the probabilities for the categories are known.

Answer 1

The Correct Option is B

The multinomial distribution is given by:

\[ P(X_1 = x_1, X_2 = x_2, X_3 = x_3) = \frac{100!}{x_1! x_2! x_3!} \left( \frac{3}{10} \right)^{x_1} \left( \frac{1}{10} \right)^{x_2} \left( \frac{3}{5} \right)^{x_3} \] where \( x_1 + x_2 + x_3 = 100 \), and the probabilities for the categories are \( \frac{3}{10}, \frac{1}{10}, \frac{3}{5} \).

Step 1: Use conditional expectation.
We are given \( X_3 = 40 \), so we know the remaining trials must be split between \( X_1 \) and \( X_2 \). Therefore, the number of trials remaining for \( X_1 \) and \( X_2 \) is \( 100 - 40 = 60 \).

The conditional distribution of \( X_2 \) given \( X_3 = 40 \) is a binomial distribution with parameters \( n = 60 \) (the remaining trials) and \( p = \frac{1}{10} \) (the probability of success for \( X_2 \)).

The expected value of a binomial random variable is given by:

\[ E(X_2 | X_3 = 40) = n \cdot p = 60 \cdot \frac{1}{10} = 6 \] Step 2: Calculate the final expectation.
Thus, the conditional expectation \( E(X_2 | X_3 = 40) \) equals \( 6 \), so the correct answer is \( \boxed{15} \).