8, Suppose that the number of customer who enter a post office on a given day is a Poisson randomvariable Z with parameter 2. Each customer who enters the post office is a male with probability p and afemale with probability 1-p. Let X denote the number of female customer entering the post office on thatgiven day, and Y denote the number of male customer entering the post office on that given day.(1) What is the probability mass function of X?(2) Is X independent of Y? Why?

Given information: The number of customers who enter the post office is denoted by Z. The number of…

Answered: 8, Suppose that the number of customer…

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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Let joint Probability distribution function of random variables X and Y be if (x, y) = (1, 1) 1/3 |1/3 if (x, y) = (0, 0) (1/3 if (x, y) = (2, 0) P(x, y)={ otherwise . Are X and Y independent?
Use the probability mass function of a uniform random variable with parameters a and b tofind its mean.
Customers arrive at a shop according to a Poisson process at a mean rate of 2 customers every ten minutes. The shop opens at 9am. (c) Let Y be the time (in hours) until the 4th customer arrives at the shop. State the distribution of Y and give the value(s) of its parameter(s). (d) Show that the probability of the time until the 4th customer arrives being greater than 15 minutes is 13e-3. (e) Let M be the number of customers who arrive at the shop between 9am and 12 noon. State the distribution of M and give the value(s) of its parameter(s). (f) Calculate P(M = 30) using a normal approximation with the continuity correction.
A random variable X has a N(0,1) distribution. Use the moment generating function of X to compute (a) the third and (b) fourth moments of X, i.e., E(X³) and E(X4).
Let (Ω, Pr) be a probability space, and let X and Y be two independent random variables that are positive and have non-zero variance. (a) Prove that X^2 and Y are independent. Note that by symmetry, it will also follow that Y^2 and X are independent. (b) Use the result from part (a) to show that the random variables W = X + Y and Z = XY are positively correlated (i.e. Cov(W, Z) > 0).
Let X(1), X(2), ..., X(n) be the order statistics of a set of n independent uniform (0, 1) random variables.Find the conditional distribution of X(n) given that X(1) = s(1), X(2) = s(2), ..., X(n-1)=s(n-1).

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