Financial Risk Manager Part 1

Explanation

For a binomial random variable X with parameters n (number of trials) and p (probability of success):

1. Expected value: E(X) = np = 2
2. Variance: Var(X) = np(1-p) = 1.5

From the variance formula: Var(X) = np(1-p) = 2(1-p) = 1.5

Solving for p: 2(1-p) = 1.5 1-p = 0.75 p = 0.25

Now, using E(X) = np = 2: n × 0.25 = 2 n = 8

So we have n = 8, p = 0.25

Calculating P(X = 5):

P(X = 5) = C(8,5) × (0.25)^5 × (0.75)^3

Where C(8,5) = 8!/(5!×3!) = 56

P(X = 5) = 56 × (0.25)^5 × (0.75)^3 = 56 × 0.0009765625 × 0.421875 = 56 × 0.0004119873046875 = 0.0230712890625 ≈ 0.023

Therefore, the correct answer is B. 0.023.