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Say we have X ~ Uniform(0, 1) and Y ~ Uniform(0, 1) and the two are independent. What is the expected value of the minimum of X and Y?
This is the same question as problem #18 in the Statistics Chapter of Ace the Data Science Interview!
Let Z = min(X, Y). Then we know the following:
For a uniform distribution, the following is true for a value of z between 0 and 1:
Since X and Y are i.i.d. (independent and identically distributed), this yields:
Now we have the cumulative distribution function for z. We can get the probability density function by taking the derivative of the CDF to obtain the following:
Then, solving for the expected value by taking the integral yields the following: