Uniform and Exponential Distribution of Random VariablesDate: 01/16/2006 at 16:49:00 From: troy Subject: Relationship between uniform distrib and exponential distrib I've seen statements claiming if you take the natural log of a uniformly distributed random variable, it becomes a exponentially distributed random variable. I know it's a true statement, but I wonder if you would provide a proof? Date: 01/17/2006 at 09:28:45 From: Doctor George Subject: Re: Relationship between uniform distrib and exponential distrib Hi Troy, Thanks for writing to Doctor Math. The standard form of the problem includes a minus sign on the logarithm, so I will include it. Let X ~ U[0,1] and Y = -ln(X). F(y) = P(Y < y) Y = P[-ln(X) < y] = P[X > e(-y)] = 1 - P[X < e(-y)] = 1 - F[e(-y)] X Now differentiate both sides with respect to y. f(y) = f[e(-y)]e(-y) = e(-y) Y X Does that make sense? Write again if you need more help. - Doctor George, The Math Forum http://mathforum.org/dr.math/ Date: 01/17/2006 at 12:26:42 From: troy Subject: Thank you (Relationship between uniform distrib and exponential distrib) Thank you, Doctor George, for the succinct proof. I'd like to continue the discussion below. Although it's not a complex derivation, it's definitely non-trivial. I wondered why most articles skipped the proof and only made the statement. Could I ask you to point me to a reference where I could find similar derivations? Also on the last equation, f(y) = f[e(-y)]e(-y) = e(-y), Y X it implies that f[e(-y)] = 1. X What is the reason for this? Is it because by definition that Y = -ln(X), therefore x = e(-y), so the probability of x = e(-y) is always 1? Thank you again for the beautiful derivation. Troy Date: 01/17/2006 at 13:58:33 From: Doctor George Subject: Re: Thank you (Relationship between uniform distrib and exponential distrib) Hi Troy, You are on the right track. Since X is uniform, f(x) = 1 X now substituting x = e(-y) we get f[e(-y)] = 1 X As for a reference, most any college level book on Mathematical Statistics will contain examples similar to this. Look for a section on transformation of variables. Sometimes a book will skip a short proof like this and have the reader work it out as a problem. - Doctor George, The Math Forum http://mathforum.org/dr.math/ |
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