Find the Probability That Two Uniform Variables Have Product Above One-Half
SquarepointData Scientist · Technical Screen
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Compute the probability that two independent uniform variables have product above one-half by conditioning or integrating over the unit square.
Let X and Y be independent random variables, each uniformly distributed on the interval (0, 1). What is the probability that X * Y > 1/2?
Constraints and Clarifications
Use the continuous uniform distribution and the given independence. Explain the relevant region or conditional probability and give an exact expression. The distinction between strict and non-strict inequality at the boundary has probability zero here.
Hints
Hint 1
Locate the feasible part of the square: For a fixed value of `X`, determine which values of `Y` can satisfy the product inequality and when that interval is nonempty.
Follow-up questions
- How does the expression change for `P(X * Y > t)` with `0 < t < 1`?
- Why is multiplying `P(X > 1/2)` and `P(Y > 1/2)` not the answer to the original question?
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