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'
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter10: Statistics
Section10.1: Measures Of Center
Problem 9PPS
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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'](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F123f1193-bc14-4498-acdc-488f1aa05f04%2F1036d96e-306a-4ed8-b990-7492a4e31240%2Fuidegyg_processed.jpeg&w=3840&q=75)
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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