Assignment # 4

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School
Simon Fraser University **We aren't endorsed by this school
Course
STAT 3083
Subject
Statistics
Date
Jun 2, 2023
Pages
1
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DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF NEW BRUNSWICK ASSIGNMENT # 4 STAT 3083 FALL 2021 Note: There will be a Quiz based on this assignment during Class on Friday, October 22, 2021. Question # 1: Let X and Y be independent random variables with E[X]=3, E[X 2 ]=25, E[Y]=10 and E[Y 2 ]=164. (a) Find E[3X + Y - 8]. (b) Find E[2X - 3Y + 7]. (c) Find Var (X). (d) Find σ x . (e) Find Var(Y). (f) Find σ y . (g) Find Var[3X + Y - 8]. (h) Find Var[2X - 3Y + 7]. (i) Find E[(X - 3)/4] and Var[(X - 3)/4]. (j) Find E[(Y - 10)/8] and Var[(Y - 10)/8]. Question # 2: A discrete random variable has moment generating function 𝑚 𝑥 (𝑡) = ? 2(𝑒 𝑡 −1) (a) Find E(X). (b) Find E(X 2 ). (c) Find σ 2 and σ . Question # 3: The probability that a wildcat well will be productive is 1/13. Assume that a group is drilling wells in various parts of the country so that the status of one well has no bearing on that of any other. Let X denote the number of wells drilled to obtain the first strike. (a) Verify that X is geometric and identify the value of the parameter p. (b) What is the exact expression for the density for X? (c) What is the exact expression for the moment generating function for X? (d) What are the numerical values of E[X], E[X 2 ], σ 2 and σ. (e) Find P[X ≥ 2] Question # 4: Let the density for X be given by ?(𝑥) = 𝑐? −𝑥 𝑥 = 1, 2, 3, ... ... ... ... (a) Find the value of c that makes this a density. (b) Find the moment generating function for X. (c) Use 𝑚 𝑥 (𝑡) to find E(X) and Var(X). Question # 5: (a) Find the moment generating function for a binomial random variable with parameters n and p. (b) Use 𝑚 𝑥 (𝑡) to show that E[X]=np. (c) Use 𝑚 𝑥 (𝑡) to show that E[X 2 ]=n 2 p 2 - np 2 + np. (d) Show that Var (X) = npq, where q = 1 - p.
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