Antiderivative of CDF Let F(x) be the CDF of a random variable. Does its antiderivative <msubsu

Jace Wright

Jace Wright

Answered question

2022-05-09

Antiderivative of CDF
Let F(x) be the CDF of a random variable. Does its antiderivative x F ( x ) d x have an interpretation?

Answer & Explanation

agentbangsterfhes2

agentbangsterfhes2

Beginner2022-05-10Added 15 answers

Step 1
Sort of. If X is an r.v. with this distribution, then this integral is equal to E ( x X ) + (where y + = m a x ( 0 , y )).
This is easy to see in the case X has a density with respect to Lebesgue measure:
x F ( y ) d y = 1 y < x ( f ( z ) 1 z < y d z ) d y = f ( z ) 1 z < y < x d y d z = f ( z ) ( x z ) + d z = E ( x X ) +
but it actually holds for arbitrary X.
Step 2
A related observation that admits a more natural interpretation is that, for a non-negative r.v. X, the antiderivative of the survival function S ( x ) = 1 F ( x ) calculated from lower bound x = 0 satisfies
where E [ Y ; A ] E [ Y 1 A ] is the restricted expectation. In particular:
0 S ( x ) d x = E [ X ]

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