Let X1, X2,. . . , Xn are independent and Xi follows Poisson(Ai) distribution, i = 1,2,... , nUse the probability generating function to show that 1X follows Poisson(i-1 Ai)distribution

As per the questionX ~ Poisson (λ)Moment Generating function is given by

Answered: Let X1, X2,. . . , Xn are independent…

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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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.
Q1) 14% of the adults in a certain population are infected by a Corona Virus. Five adults are selected randomly from this population for diagnoses. Find the probability that at least two of them will be infected by .this Virus 1 .Q2) Suppose that f(X)=X,12). a) Find .b) Find the mean Q3) A study shows that the systolic blood pressures for adults in a certain population are approximately normally distributed with mean of 115 and .standard deviation of 10 a) Find the probability that a randomly selected person will have a blood -pressure between 109 and 124 Sel).b) Find the reading that is exceeded by only 5% of the population التي يتجاوزها 5% من المجتمع(
Findi) the joint probability function of ?1and ?2.ii) the probability that less than 3 heads will occur and the student will win R1 or less.
There are two boxes. In each box there are 4 cards with a different number printed on it. The four cards have been numbered as 1,2,3,4 in each box. Two cards are drawn random from each box. The random variable X represents the difference between the number on the card from box 1 minus the number on the card from box 2.a) Find the mean of this probability distribution. (i.e. Find E[X] )b) Find the variance and standard deviation of this probability distribution.(i.e. Find V[X] and SD[X])The random variables M and W are defined as follows:M = X+5 and W = (1/2)X+5c) Find E[M] and E[W].d) Find V[M] and V[W].e) Mary and William play a game using the cards in the above boxes. Randomly two cards are drawn, and Mary records his score using the random variable M and William uses the random variable W. They repeat this for a large number of times and compare their scores. Comment on any likely differences or similarities of their scores.
HH=2, JJ=1, LL=3, NN=fifth (5th)
(b) Suppose the number of customers at a Sushi outlet is Poisson distributed where an average of nineteen customers are served per 30 minutes. (i) Over a fifteen-minute period, how many customers are served on average? Find the probability of exactly seven customers served over a fifteen-minute period. Write your answer correct to 4 decimal places. (iii) Calculate the probability at least 1 customer is served over a fifteen-minute period. Write your answer correct to 4 decimal places.
The flow of goods through a certain machine in a factory can be modeled as a Poisson Random Process. Suppose that a machine has 20 goods per minute as an outflow rate, determine the probability that the 7th good will exit within 30 seconds. (Express your answer in 4 decimal places)
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The time between arrivals of customers at an automatic teller machine (ATM) is an exponential random variable with a mean of five minutes. a) What is the probability that the time until the third customer arrives in more than 10 minutes? b) what is the value for σ2 for the third customer? c) what is the value for σσ for the third customer ?

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Let X1, X2,. . . , Xn are independent and Xi follows Poisson(Ai) distribution, i = 1,2,... , n Use the probability generating function to show that 1X follows Poisson(i-1 Ai) distribution