Puzzle: random size confetti

Today’s puzzle is about statistical sampling, adapted from a problem I found in this book.

Professor X teaches a probability class. He assigns a holiday-themed project to his students.

Each student is to create a 500 rectangular-shaped confetti pieces, with length and width to be random numbers between 0 and 1 inches.

Alice goes home and gets started. She interprets the assignment as follows. Alice generates two random numbers from the uniform distribution, and then she uses the first number as the length and the second as the width of the rectangle.

Bob interprets the assignment differently. He instead generates one random number from the uniform distribution, and he uses that number for both the length and width, meaning he creates squares of confetti.

Clearly Alice and Bob will cut out different shapes of confetti. But how will the average size of the confetti compare?

That is, will the average area of the shapes that Alice and Bob cut out be the same? If not, whose confetti will have a larger average area?

As usual, I have posted a solution in the comments section.

Can you solve it?



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  • http://www.mindyourdecisions.com/blog/ Presh Talwalkar

    Answer to the puzzle

    Let X be a random variable with a uniform distribution.

    Bob takes one realization of X, so the area he cuts out will be distributed as X2, and the expected area is E(X2)

    Alice instead takes two realization of X. The area she cuts out will be E(X)*E(X), or E2(X).

    The difference between Bob’s expected area and Alice’s is:

    E(X2) – E2(X) = Var(X) >= 0

    The difference between Bob’s expected areas and Alice’s is equal to the variance, which is always non-negative. Notice this formula holds for random variables of other distributions too, like normal distributions or discrete distributions.

    In the case of the uniform distribution from 0 to 1, the variance is 1/12.

    So Bob’s areas will always be at least as large or larger than Alice’s. So Bob may need a little bit more paper than Alice when cutting his confetti.

  • jsn

    Bob’s area on average = 0.333… inches^2
    Alice’s area on average = 0.25 inches^2

  • Leipaella

    Bob’s areas won’t *always* be larger or as large as Alices. But on average they will be, and their expected size is.





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