Question

Let \( \{X_n\}_{n \geq 1} \) be a sequence of independent and identically distributed random variables each having uniform distribution on [0, 3]. Let \( Y \) be a random variable, independent of \( \{X_n\}_{n \geq 1} \), having probability mass function \[ P(Y = k) = \frac{1}{(e - 1) k!}, \quad k = 1, 2, \dots \] Then \[ P(\max(X_1, X_2, \dots, X_Y) \leq 1) = _________ \text{ (round off to 2 decimal places).} \]

Hint:

For problems involving the maximum of a random sample, use the distribution of the maximum and calculate the desired probabilities accordingly.

Answer 1

Correct Answer: 0.2

Using the given information, we calculate the probability that the maximum of the sequence \( X_1, X_2, \dots, X_Y \) is less than or equal to 1. The probability is: \[ P(\max(X_1, X_2, \dots, X_Y) \leq 1) \approx 0.21. \] Thus, the value is \( 0.21 \).