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Answer: Continuous uniform with parameters 0 and 1,300
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.
Author: Nikitesh Somanthe
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The amount of profit (X) for a sales company is continuously distributed uniformly with the parameters 0 and 1,500. However, a financial analyst believes that the actual profit (Y) is a minimum of X. What is the conditional distribution of X given X<1,300?
A
Continuous uniform with parameters 0 and 1,300
B
Continuous uniform with parameters 0 and 1,500
C
Continuous uniform with parameters 0 and 1,000
D
Continuous uniform with parameters 0 and 2,800