Financial Risk Manager Part 1

Explanation

Given:

• $X \sim U(0, 1500)$
• $Y = \min(X, 1300)$
• We need the conditional distribution of $X$ given $X < 1300$

Step 1: Conditional Probability Calculation

The conditional distribution is given by: $P(X < x | X < 1300) = \frac{P(X < x, X < 1300)}{P(X < 1300)}$

Step 2: Simplify the Numerator

Since $x < 1300$ in the conditional context: $P(X < x, X < 1300) = P(X < x)$

Step 3: Calculate Probabilities

For uniform distribution $U(0, 1500)$:

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

Step 4: Final Conditional Probability

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

Step 5: Interpretation

The result $\frac{x}{1300}$ represents the cumulative distribution function of a uniform distribution $U(0, 1300)$. Therefore, the conditional distribution of $X$ given $X < 1300$ is:

Continuous uniform with parameters 0 and 1,300

This makes intuitive sense because when we condition on $X < 1300$, we are essentially restricting our sample space to the interval [0, 1300], and the uniform distribution property is preserved within this restricted range.