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(1−F(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(1−F(x))|∞0−∫∞0x(−ddxF(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 limx→∞1−F(x)1x=limx→∞x2f(x)
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