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If f is a pdf of a real-valued random variable X, the cumulative distribution func- tion (cdf) of X is F : R→ [0, 1] so that F(x) = √∞f(t) dt for all x. For an event AC R, let 1[A] be 1 if A is true and 0 otherwise. Let X1,..., Xn be n samples of X that are independent and identically distributed with cdf F. For all x, define Ên(x) = ½-½ Σï²±₁₁[X; ≤ x] known as an empirical estimate of F(x). n n i=1 (a) For any x Є R, 1[X ≤ x] is 1 when X ≤ x and 0 otherwise. Show that F(x) = E[1[X ≤ x]]. (b) For any x and n, what is E[Fn(x)]? (c) For any x, show that the variance of Ĥ₁(x) is Var (Ĥ₁(x)) = F(x)(1 − F(x)). (d) For any x and n, show that the variance of Fn(x) is 1½-1 F(x)(1 − F(x)). (e) For any x and n, show that E[(Fn(x) − F(x))²] ≤ 4n'

If f is a pdf of a real-valued random variable X, the cumulative distribution func- tion (cdf) of X is F : R→ [0, 1] so that F(x) = √∞f(t) dt for all x. For an event AC R, let 1[A] be 1 if A is true and 0 otherwise. Let X1,..., Xn be n samples of X that are independent and identically distributed with cdf F. For all x, define Ên(x) = ½-½ Σï²±₁₁[X; ≤ x] known as an empirical estimate of F(x). n n i=1 (a) For any x Є R, 1[X ≤ x] is 1 when X ≤ x and 0 otherwise. Show that F(x) = E[1[X ≤ x]]. (b) For any x and n, what is E[Fn(x)]? (c) For any x, show that the variance of Ĥ₁(x) is Var (Ĥ₁(x)) = F(x)(1 − F(x)). (d) For any x and n, show that the variance of Fn(x) is 1½-1 F(x)(1 − F(x)). (e) For any x and n, show that E[(Fn(x) − F(x))²] ≤ 4n'

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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If f is a pdf of a real-valued random variable X, the cumulative distribution func- tion (cdf) of X is F : R→ [0, 1] so that F(x) = √∞f(t) dt for all x. For an event AC R, let 1[A] be 1 if A is true and 0 otherwise. Let X1,..., Xn be n samples of X that are independent and identically distributed with cdf F. For all x, define Ên(x) = ½-½ Σï²±₁₁[X; ≤ x] known as an empirical estimate of F(x). n n i=1 (a) For any x Є R, 1[X ≤ x] is 1 when X ≤ x and 0 otherwise. Show that F(x) = E[1[X ≤ x]]. (b) For any x and n, what is E[Fn(x)]? (c) For any x, show that the variance of Ĥ₁(x) is Var (Ĥ₁(x)) = F(x)(1 − F(x)). (d) For any x and n, show that the variance of Fn(x) is 1½-1 F(x)(1 − F(x)). (e) For any x and n, show that E[(Fn(x) − F(x))²] ≤ 4n'
Transcribed Image Text:If f is a pdf of a real-valued random variable X, the cumulative distribution func- tion (cdf) of X is F : R→ [0, 1] so that F(x) = √∞f(t) dt for all x. For an event AC R, let 1[A] be 1 if A is true and 0 otherwise. Let X1,..., Xn be n samples of X that are independent and identically distributed with cdf F. For all x, define Ên(x) = ½-½ Σï²±₁₁[X; ≤ x] known as an empirical estimate of F(x). n n i=1 (a) For any x Є R, 1[X ≤ x] is 1 when X ≤ x and 0 otherwise. Show that F(x) = E[1[X ≤ x]]. (b) For any x and n, what is E[Fn(x)]? (c) For any x, show that the variance of Ĥ₁(x) is Var (Ĥ₁(x)) = F(x)(1 − F(x)). (d) For any x and n, show that the variance of Fn(x) is 1½-1 F(x)(1 − F(x)). (e) For any x and n, show that E[(Fn(x) − F(x))²] ≤ 4n'
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