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.

For a uniform distribution U(a,b), the cumulative distribution function (CDF) is:

$F(x) = \frac{x - a}{b - a} \quad \text{for } a \leq x \leq b$

For X ~ U(-1,5):

• a = -1
• b = 5
• b - a = 5 - (-1) = 6

So the CDF is:

$F(x) = \frac{x - (-1)}{6} = \frac{x + 1}{6}$

To generate X from U ~ U(0,1), we solve for x in terms of u:

$u = \frac{x + 1}{6}$

$x + 1 = 6u$

$x = 6u - 1$

Now, for u = 0.10:

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

Therefore, the corresponding random variable from U(-1,5) is -0.40.

Verification:

• When u = 0, x = 6×0 - 1 = -1 (lower bound)
• When u = 1, x = 6×1 - 1 = 5 (upper bound)
• When u = 0.10, x = -0.40 (correct)