GATE DA 2024 Set 1 — Question 56
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Probability & Statistics → Continuous Random Variables → Independent RVs (Continuous)
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Question
Let be a random variable uniformly distributed in the interval and be a random variable uniformly distributed in the interval . If and are independent of each other, the probability is ______ (rounded off to three decimal places).
Correct answer
0.125 to 0.125
Solution
Given:
The condition implies .
Considering the support of and :
The total area of the support rectangle is .
The favorable region is the triangle with vertices .
Base of triangle (along ) is .
Height of triangle (at ) is .
Area of favorable region = .
Probability = .
- , so PDF for .
- , so PDF for .
- and are independent, so the joint PDF is over the rectangular region .
The condition implies .
Considering the support of and :
- Minimum is 2. For to be possible, must be at least 2.
- So we look at the region where and .
The total area of the support rectangle is .
The favorable region is the triangle with vertices .
Base of triangle (along ) is .
Height of triangle (at ) is .
Area of favorable region = .
Probability = .
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