Financial Risk Manager Part 1

Explanation

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

• Expected value: E(X) = np
• Variance: Var(X) = np(1-p)

Step 1: Use the given probabilities

We are given:

• P(X=0) = 0.20
• P(X=1) = 0.35
• E(X) = 1.5

For a binomial distribution:

• P(X=0) = (1-p)^n = 0.20
• P(X=1) = n × p × (1-p)^(n-1) = 0.35

Step 2: Use the ratio of probabilities

Divide P(X=1) by P(X=0):

$$\frac{P(X=1)}{P(X=0)} = \frac{n \times p \times (1-p)^{n-1}}{(1-p)^n} = \frac{n \times p}{1-p} = \frac{0.35}{0.20} = 1.75$$

So: $\frac{n \times p}{1-p} = 1.75$

Step 3: Use the expected value

We know E(X) = np = 1.5

Substitute np = 1.5 into the ratio: $\frac{1.5}{1-p} = 1.75$ $1.5 = 1.75(1-p)$ $1.5 = 1.75 - 1.75p$ $1.75p = 1.75 - 1.5$ $1.75p = 0.25$ $p = \frac{0.25}{1.75} = \frac{1}{7} \approx 0.142857$

Step 4: Find n

Since np = 1.5 and p = 1/7: $n \times \frac{1}{7} = 1.5$ $n = 1.5 \times 7 = 10.5$

Since n must be an integer, this suggests the distribution may not be exactly binomial, but we can proceed with the variance calculation.

Step 5: Calculate variance

Var(X) = np(1-p) = 1.5 × (1 - 1/7) = 1.5 × (6/7) = 9/7 ≈ 1.2857

This rounds to 1.3, which matches option D.

Answer: D (1.3)