Example 1.4 Let X1, X2,... be independent with common density ax-a-1, for x > 1, a > 0, f(x) = 0, otherwise, and set Y = n/a. max1 1, F(x)= 0, from which it follows that, for any x > 0, otherwise, FY, (x) = P(max Xk ≤ xn¹/a) = (F(xn¹/a))" 1≤k≤n = (1-1)" as no.
Example 1.4 Let X1, X2,... be independent with common density ax-a-1, for x > 1, a > 0, f(x) = 0, otherwise, and set Y = n/a. max1 1, F(x)= 0, from which it follows that, for any x > 0, otherwise, FY, (x) = P(max Xk ≤ xn¹/a) = (F(xn¹/a))" 1≤k≤n = (1-1)" as no.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
Related questions
Question
Transcribed Image Text:Example 1.4 Let X1, X2,... be independent with common density
ax-a-1, for x > 1, a > 0,
f(x) =
0,
otherwise,
and set Y = n/a. max1<k≤n Xk, n ≥ 1. Show that Y converges in distribution
as noo, and determine the limit distribution.
In order to solve this problem, we first compute the distribution function:
dy 1 xa, for x > 1,
F(x)=
0,
from which it follows that, for any x > 0,
otherwise,
FY, (x) = P(max Xk ≤ xn¹/a) = (F(xn¹/a))"
1≤k≤n
= (1-1)"
as no.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 6 images
Recommended textbooks for you
A First Course in Probability (10th Edition)
Probability
ISBN:
9780134753119
Author:
Sheldon Ross
Publisher:
PEARSON
A First Course in Probability (10th Edition)
Probability
ISBN:
9780134753119
Author:
Sheldon Ross
Publisher:
PEARSON