Let X₁, X₂ be independent and identically distributed random variables following a geometricdistribution with parameter p € (0, 1), that is, P(X₁ = k) = pk-1(1-p) for k € N.a) Derive a formula for P(X₁2 k) for ke N.b) Determine the cumulative distribution function of Y = min(X₁, X₂).c) Let Z= max(X₁, X₂). Are Y and Z independent? Justify your answer.

Given that X1, X2 be independent and identically distributed random variables following a geometric…

Answered: Let X₁, X₂ be independent and…

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...

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Show that if X and Y are independent Exp(a)-distributed random vari- ables, then X/Y E F(2,2).
The probability mass function of a discrete random variable is: 7-x p(x) = a=) G) for x = 1,2,., 14 (a) Compute Pr{35) (c) Compute the expected value, the variance, and the standard deviation.
a) Let X₁, X2, X3,..., X, be a random sample of size n from population X. Suppose that X~N(0, 1) ΣΧι – θνη. √n and Y = i) Show that the standard score of the sample mean X, is equal to Y. ii) Show that the mean and variance of the random variable Y are 0 and 1, respectively. iii) Show using the moment generating function technique that Y is a standard normal random variable. iv) What is the probability that Y² is between 0.02 and 5.02?
Suppose X - Fm,n- a) Find the distribution of 1/X. b) Find the distribution of (m/n)X Y = 1+ (m/n)X' Identify the distribution with appropriate parameters.
You opened a 3d printing start up, by acquiring 10 used industrial 3d printers. The previous owner informed you that 8 are working properly with only a 10% chance of producing noticeable defects and 2 that became defective resulting a 20% chance of producing noticeable defects. However, he forgot to label which printers are defective. (a) To test for a defective printer, you randomly select a printer and print 5 structures. If 2 of the 5 printed items is defectively printed, what is the probability that the chosen printer is defective? (b) What is the probability that any given item printed by your start up isdefective? (Assume that you do not know which machines are defective)
Let Y be a Poisson random variable with mean λ = 2. (a) Find P(Y ≥ 2). (b) Find P(Y ≥ 4|Y ≥ 2).
a) Let X₁, X₂, X3,..., X, be a random sample of size n from population X. Suppose that X~N(0, 1) you and Y = -√n. VR i) Show that the standard score of the sample mean X, is equal to Y. ii) Show that the mean and variance of the random variable Y are 0 and 1, respectively. iii) Show using the moment generating function technique that Y is a standard normal rando... variable. iv) What is the probability that Y² is between 0.02 and 5.02? b) Let X₁, X₂, X, and Y₁, Y₂.... Y be random samples from populations with moment generating functions Mx, (t) = eat+t² and My (t) = (-), respectively. 25 i) Find the sampling distribution of the statistic W = X₁ + 2X₂ X3 + X₁ + X5. ii) What is the value of the sample size n, if PIX(X-X)2> 68.3392] = 0.025? iii) What is the value of the sample size m, if P(|Y-Hy ≥10) <0.04? QUESTION A
Assume that X is a random variable whose conditional distribution given the variable Y is poisson P (X | Y) = Po (Y). Suppose further that Y has a gamma distribution Y ∼ Gamma (1, 1). (a) Determine the value E (XY). (b) Determine the conditional distribution P (Y | X).
Let X1,...,Xn be iid random variables with expected value 0, variance 1, and covariance Cov [Xi,Xj] = ρ, for i≠j. Use Theorem of linearity of expectation to find the expected value and variance of the sum Y = X1 +...+Xn.
a) Let X₁, X₂, X₁,...,X, be a random sample of size n from population X. Suppose that X-N(0, 1) and Y = -1-8√n. i) Show that the standard score of the sample mean X, is equal to Y. ii) Show that the mean and variance of the random variable Y are 0 and 1, respectively. iii) Show using the moment generating function technique that Y is a standard normal random variable. iv) What is the probability that Y2 is between 0.02 and 5.02?
Let Y₁, Y2, Y3, Y₁, Y5 be a random sample of size 5 from a standard normal population. Find the moment generating function of the statistic: X = 2Y₁² + Y₂² + 3Y3²+ Y₂² +4Y5² Let Y₁, Y₂, Y3, Y4, Y5 and X₁, X2,..., X, be independent and normally distributed random samples from populations with means ₁ = 2 and ₂ = 8 and variances ₁² = 5 and ₂² = k₁ respectively. Suppose that P(X-Y> 10) = 0.02275, find the value of 0₂² = k.

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