Sunday, February 14, 2016

limits - What is limxtoinftyx2f(x) where inti0nftyf(x),mathrmdx=1

I encountered a question in a test:




If F(x) is a cumulative distribution function of a continuous non-negative random variable X, then find 0(1F(x))dx



After a bit of pondering, I thought that the answer should depend upon the density of the random variable, so I checked the "none of these" option , but the correct answer was E(X).So later I tried to work out the question properly.


If the density of random variable X is f(x) then it is necessary that f(x)>0 and 0f(x)dx=1 Doing the integration by parts x(1F(x))|00x(ddxF(x))dx

which reduces to limxx(1F(x))+0xf(x)dx


Now the 0xf(x)dx is clearly E(X) but the limit is where my problem arises. Applying L'Hopital rule in the limit we have limx1F(x)1x=limxx2f(x)

. Now is there any way to further reduce that limit to 0 so that E(X) is the correct answer or am I doing something wrong?

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