Financial Risk Manager Part 1

Explanation

To generate random variables from U(-1,5) using random variables from U(0,1), we use the inverse transform method.

Step 1: Find the CDF of X ~ U(-1,5)

For a uniform distribution U(a,b), the CDF is: $F(x) = \frac{x - a}{b - a}, \quad \text{for} \quad a \leq x \leq b$

Given X ~ U(-1,5), where a = -1 and b = 5: $F(x) = \frac{x - (-1)}{5 - (-1)} = \frac{x + 1}{6}$

Step 2: Apply the inverse transform method

Let U ~ U(0,1), then we solve for X: $U = F(X) = \frac{X + 1}{6}$ $X = 6U - 1$

Step 3: Calculate for U = 0.10

$X = 6 \times 0.10 - 1 = 0.60 - 1 = -0.40$

Therefore, the corresponding random variable for U = 0.10 is -0.40, which corresponds to option C.

Verification

The transformation X = 6U - 1 maps:

• When U = 0 → X = -1 (lower bound)
• When U = 1 → X = 5 (upper bound)
• When U = 0.10 → X = -0.40 (as calculated)