Answer: Continuous uniform with parameters 0 and 1,300.

## 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.

Author: Tanishq Prabhu