## Interview Question

Summer Analyst - IBD Strat Interview

-New York, NY

# If two random variables have a uniform probability distribution between 0 and 1, what is the probability that their product is less than 0.5?

4

here is a nice blog post summarizing that very same question. http://bayesianthink.blogspot.com/2012/11/fun-with-uniform-random-numbers.html there are a bunch of other interview questions too.

matrix winter on

7

The answer is 1/2 + 1/2 * ln(2) and the analytical derivation is trivial. All you need to know is how to integrate dx/x.

bbzippo on

1

The answer with ln(2) seemed firstly misleading but actually it coincided with mine. After drawing an x-y plane, plotting y=1/(2x) and calculating the area under the graph, minding constraints of the uniformly distributed variables (0,1). Which is exactly 1*(1/2) + integral(1/2x) from {1/2,2}.

Anonymous on

1

This probability cannot be calculated analytically, or at least not by using the transformation U = X1*X2, V = X2, marginalizing out V, and integrating the resulting density for U from 0 to .5. You can evaluate the integrals numerically though, yielding the result: 0.8465. I think that all the interviewers desired was to say that if either random variable is less than 0.5, certainly the product will be less than 0.5, so we know the probability that the result is less than 0.5 is at least 0.75.

Tophat on