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(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
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 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 ... 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 ... 

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.
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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