Let X be uniform random variable on [0, 4] and Y be uniform random variable on [0, 1]. If X and Y are independent, then P(max{X, Y} > 3) is equal to:

The correct answer is ption 1: 1/4

Key Points

We are given:

• X ~ Uniform(0, 4)
• Y ~ Uniform(0, 1)
• X and Y are independent

We are asked to find: \(P(\max\{X, Y\} > 3)\)

Note: \(\max(X, Y) \leq 3 \Rightarrow X \leq 3 \text{ and } Y \leq 3\)

But since Y ~ Uniform(0, 1), it always satisfies Y ≤ 3. So, the event \(\max(X, Y) \leq 3\) is equivalent to \(X \leq 3\)

So,

\(P(\max(X, Y) > 3) = 1 - P(\max(X, Y) \leq 3) = 1 - P(X \leq 3)\)

Since X ~ Uniform(0, 4),

\(P(X \leq 3) = \frac{3 - 0}{4 - 0} = \frac{3}{4}\)

Therefore,

\(P(\max(X, Y) > 3) = 1 - \frac{3}{4} = \frac{1}{4}\)

Correction: The correct value is 1/4, so the correct answer is:

option 1: 1/4

Additional Information

• The maximum of two variables exceeds a threshold if at least one exceeds it.
• Since Y ∈ [0,1], and 1 < 3, the only contributing factor is whether X > 3.
• P(X > 3) = (4 − 3) / (4 − 0) = 1/4

Hence, the correct answer is: option 1: 1/4