Let 𝑋 be a random variable uniformly distributed in the interval [1, 3] and…
2024
Let 𝑋 be a random variable uniformly distributed in the interval [1, 3] and 𝑌 be a random variable uniformly distributed in the interval [2, 4]. If X and Y are independent of each other, the probability P(𝑋 ≥ 𝑌) is ______ (rounded off to three decimal places).
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Correct answer: 0.125
Answer: 0.125 (rounded to three decimal places)
Key idea: Use the joint uniform distribution over the rectangle [1,3] × [2,4] and compute the area where X ≥ Y.
The joint pdf is constant on the rectangle of area 2 × 2 = 4, so the density is 1/4.
The region where X ≥ Y inside the rectangle is given by points with 1 ≤ X ≤ 3, 2 ≤ Y ≤ 4, and X ≥ Y. This region exists only for 2 ≤ Y ≤ 3; for Y > 3 there are no X ≤ 3 satisfying X ≥ Y.
For 2 ≤ Y ≤ 3, X runs from X = Y to X = 3, so the area of the favorable region is ∫_{y=2}^{3} (3 − y) dy = 0.5.
Multiply area by density: probability = 0.5 × (1/4) = 0.125.
Final answer: 0.125