(a) Consider the random variable Y(1) min{Y1,..., Y,n}. Verify that the density function for Y(1) is fY, (2) = n(1 – 2)"-1 provided 0 < z < 1. (b) Consider the random variable Xn = nY1).Verify that the density function for Xn is = (1 –)". X\ n-1 fx,(x) n for 0 n, Fx, (x) = 1 0, x < 0. (c) Verify that - e-*, x > 0, lim Fx,(x) : x < 0. This proves Xn converges in distribution to an Exp(1) random variable. Note that this calculation uses one of the definitions from first-year calculus of the number e as a certain limit. You might need to look that up.
(a) Consider the random variable Y(1) min{Y1,..., Y,n}. Verify that the density function for Y(1) is fY, (2) = n(1 – 2)"-1 provided 0 < z < 1. (b) Consider the random variable Xn = nY1).Verify that the density function for Xn is = (1 –)". X\ n-1 fx,(x) n for 0 n, Fx, (x) = 1 0, x < 0. (c) Verify that - e-*, x > 0, lim Fx,(x) : x < 0. This proves Xn converges in distribution to an Exp(1) random variable. Note that this calculation uses one of the definitions from first-year calculus of the number e as a certain limit. You might need to look that up.
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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Suppose that Y1,Y2,..., are independent and identically distributed Unif(0,1) random variables so that their common density
a). Consider the random variable Y(1) = min{Y1 , . . . , Yn }. Verify that the density function for Y(1) is ... provided 0 ≤ z ≤ 1.
b). Consider the random variable Xn = nY(1).Verify that the density function for Xn is ... for 0 ≤ x ≤ n. This implies that the distribution function for Xn is ...
c). Verify that ...
Transcribed Image Text:5.
Suppose that Y1, Y2,..., are independent and identically distributed Unif(0, 1) random
variables so that their common density function is fy(y) = 1 for 0 < y< 1.
(a) Consider the random variable Y(1) = min{Y1,..., Yn}. Verify that the density function for
Y(1) is
fY (2) = n(1 – 2)"-1
provided 0 < z < 1.
(b) Consider the random variable Xn = nY(1). Verify that the density function for Xn is
fx. (a) = (1- )".
п-1
fx, (x) = (1
for 0< x < n. This implies that the distribution function for Xn is
1,
x > n,
(1-)"
n
Fx,(x) =
1
0 <x < n,
0,
x < 0.
6,
(c) Verify that
x > 0,
lim Fx,(x) =
x < 0.
This proves Xn converges in distribution to an Exp(1) random variable. Note that this
calculation uses one of the definitions from first-year calculus of the number e as a certain
limit. You might need to look that up.
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