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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
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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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.
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}.
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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