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Unformatted text preview: 550.420 Introduction to Probability Homework #10
Due November .LO) 2011 1. (2 points) Let. the joint distribution function of X , Y. and Z be given by Rg_y'!z(;17,y. z) = (1 — 9“”)(1 — €42le — EA“) for .131 ,2 > 0. (a) Are X. Y. and Z independent?
(b) Find the joint density function of X, Y. and Z.
(c) Find P[X < Y < Z]. 2. (2 points) Let X1. X2, ..., Xn be a random sample of size n. from a
continuous distribution function F with density function f. (a) Calculate the density function of the sample range R = X0.) — X“). (b) Use (a) to find the density function of the range of 11 random numbers
from (0.1). 3. (2 points} Let X 1. X2. X” be independent random variables from a
Uniform((), 6) distribution. where 9 > 0. Show that. n — 1
ElR] : 71+ 19' where R is the range of the sample of random variables. 1 Hint: Use part. (a) of problem 2. n+1 1H R as an estimator of 6 when it. is Remark: This is the basis for using
unknown. 4. (2 points) Let X. Y. and Z be independent. random variables, each with
mean 0 and variance 1. Calculate 13pm}! + 52V]. 5. (2 points) A company puts ﬁve different. types of prizes into their cereal
boxes‘ one in each box and in equal proportions. If a customer decides to
collect all ﬁve prizes, what is the expected number of the boxes of cereals
that he or she will buy? ...
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- Fall '11