1. (a) Let X be a continuous, non-negative (that is, P(X ≥ 0) = 1) random variable with PDF f. Show that E(X) = ™P(X > x)dx. Hint: Note that P(X > x) = f f(y)dy. Substitute this into fo P(X> x)dx, and then switch the order of integration. Note: This is an analogue of the Tail Sum Formula for Expectation, proved in class for non-negative, integer-valued random variables.
1. (a) Let X be a continuous, non-negative (that is, P(X ≥ 0) = 1) random variable with PDF f. Show that E(X) = ™P(X > x)dx. Hint: Note that P(X > x) = f f(y)dy. Substitute this into fo P(X> x)dx, and then switch the order of integration. Note: This is an analogue of the Tail Sum Formula for Expectation, proved in class for non-negative, integer-valued random variables.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Transcribed Image Text:1. (a) Let X be a continuous, non-negative (that is, P(X ≥ 0) = 1) random
variable with PDF f. Show that
E(X) =
™P(X > x)dx.
Hint: Note that P(X > x) = f f(y)dy. Substitute this into fo P(X>
x)dx, and then switch the order of integration.
Note: This is an analogue of the Tail Sum Formula for Expectation,
proved in class for non-negative, integer-valued random variables.
![(b) Suppose that X is a continuous random variable with PDF f. Let g be a
deterministic, non-negative function. Prove the Law of the Unconscious
Statistician (in the special case that g is non-negative)
E[g(x)] =
g(x)f(x)dx
using part (a).
Hint: Note that P(g(X) > y) = fµ‚ ƒ(x)dx, where Ay = {x : g(x) > y}.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa54c3a1b-c0c6-4618-b897-4acbc63b0ce8%2F061fa6c6-4053-4477-b757-93f67643262b%2Fitl5o6_processed.png&w=3840&q=75)
Transcribed Image Text:(b) Suppose that X is a continuous random variable with PDF f. Let g be a
deterministic, non-negative function. Prove the Law of the Unconscious
Statistician (in the special case that g is non-negative)
E[g(x)] =
g(x)f(x)dx
using part (a).
Hint: Note that P(g(X) > y) = fµ‚ ƒ(x)dx, where Ay = {x : g(x) > y}.
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