Financial Risk Manager Part 1

The conditional distribution of X given X < 1,300 is derived as follows:

Given: X ~ U(0, 1500) and Y = min(X, 1300)

We need P(X < x | X < 1300):

$P(X < x | X < 1300) = \frac{P(X < x, X < 1300)}{P(X < 1300)}$

Since x < 1300, P(X < x, X < 1300) = P(X < x)

$= \frac{P(X < x)}{P(X < 1300)} = \frac{\frac{x}{1500}}{\frac{1300}{1500}} = \frac{x}{1300}$

This shows that the conditional distribution function is $\frac{x}{1300}$ for 0 ≤ x ≤ 1300, which corresponds to a continuous uniform distribution with parameters 0 and 1,300.

Intuitively, when we condition on X < 1,300, we're restricting the original uniform distribution (0, 1500) to the interval (0, 1300), which remains uniform over that smaller interval.