Question

Let X be a nonnegative random variable with density function f. Show that if r(u) = ?_u^? f(t) dt, then E(X) = ?_0^? P(X ? u) du = ?_0^? r(u) du. [Hint: this is the analogue of No. 18. Calculation with integrals is smoother than with sums.]

          Let X be a nonnegative random variable with density function f. Show that if r(u) = ?_u^? f(t) dt, then
E(X) = ?_0^? P(X ? u) du = ?_0^? r(u) du.
[Hint: this is the analogue of No. 18. Calculation with integrals is smoother than with sums.]

Let X be a nonnegative random variable with density function f. Show that if r(u) = ?u^? f(t) dt, then
E(X) = ?0^? P(X ? u) du = ?0^? r(u) du.
[Hint: this is the analogue of No. 18. Calculation with integrals is smoother than with sums.]

Added by Tyler L.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

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Solved by Expert Madhur L

12/15/2022

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Let X be a nonnegative random variable with density function f . Show that if r(u) = ∫_u^∞ f(t) dt, then E(X) = ∫_0^∞ P(X ≥ u) du = ∫_0^∞ r(u) du. [Hint: this is the analogue of No. 18. Calculation with integrals is smoother than with sums.]