Q 4.4. An individual picked at random from a population has a propensity to have accidentsthat is modelled by a random variable Y having the gamma distribution with shape parameter aand rate parameter ß. Given Y = y, the number of accidents that the individual suffers in years1, 2, ..., n are independent random variables X₁, X2,... Xn each having the Poisson distributionwith parameter y.(a) Write down a function f so that the joint distribution of Y, X₁, Xn can be describedviaP(a ≤ Y ≤ b, X₁ = k1, X₂ = k2 … . . Xn = kn) = [° ƒ (y, k1, k2, ... kn)dyand derive from this expression that, for your choice of f, Y has the Gamma distribution,and that conditionally on Y = y, X₁, X2,... Xn are independent, each having the Poissondistribution with parameter y. (b) Find the conditional distribution of Y given that X₁ = k₁, X2 = k2, ..., kn.(c) An insurance company has observed the number of accidents that an individual has sufferedon each of n years and wishes to predict the number of accidents that individual willexperience in the next year. To this end let Xn+1 be a further random variable so that,conditionally on Y y, X1, X2,... Xn, Xn+1 are independent, each having the Poissondistribution with parameter y. Write down the value of the conditional expectationE[Xn+1 X1, X2, ... Xn, Y],-and hence determineE[Xn+1|X1, X2, ... Xn].

Given that the random variable Y has the gamma distribution with shape parameter and rate parameter…

Answered: Q 4.4. An individual picked at random…

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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A certain mutual fund invests in both U.S. and foreign markets. Let x a random variable that represents the monthly percentage return for the fund. Assume x has mean u = 1.9% and standard deviation o = 0.3%. (a) The fund has over 175 stocks that combine together to give the overall monthly percentage return x. We can consider the monthly return of the stocks in the fund to be a sample from the population of monthly returns of all world stocks. Then we see that the overall monthly return x for the fund is itself an average return computed using all 175 stocks in the fund. Why would this indicate that x has an approximately normal distribution? Explain. Hint: See the discussion after Theorem 6.2. The random variable -Select-v is a mean of a sample size n = 175. By the -Select- v, the -Select- distribution is approximately normal. (b) After 6 months, what is the probability that thẹ average monthly percentage return x will be between 1% and 2%? Hint: See Theorem 6.1, and assume that x has…
12. i. State Central limit theorem and tell what is the main theme of it. ii. Show that if the random variables (X, Y) follow the probability density of a bivariate normal distribution and if the correlation p = 0, X and Y are independent. (Show that f(x, y) = g(x) h(y))
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Suppose that monthly income follows a normal distribution with mu=$4.400 and sigma=$1,600 A government agency wants to provide free health care for all individuals who fall below a specific monthly income level but does not want to award this benefit to more than 33% of the population What is the income catoff for this proposed benefit where no more than 33% of the population will qualify (in other words , will fall below this level of income ?
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certain city, the daily consumption of water (in millions of liters) follows approximately a gamma distribution with rate = 0.5 and shape parameter=4. If the daily capacity of that city is 8 million liters of water, what is the probability that on any given day the water supply is more than the daily capacity? O 0.174 O 0.5665 O 0434 O 0.8264
34. Suppose that X is a random variable with MGF M(t) = (1/8)e + (1/4)e + (5/8)e. (a) What is the distribution of X? (b) What is P[X= 2]?
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3.4 Solve the below problem: Let Y, and Y, have a bivariate normal distribution: 2. 1. 2. Y2-H2 exp 2no,02y1-p2 2(1-p2) {. (ज-र.) + ( ज-र) (ज-पदर) 07 (जब)} Dr= –00 < y, < ∞, -00 < y2 < ∞, Show that the marginal distribution of Y, is normal with mean µz and variance o is given by: 2, 1 f(y2) = 203 ,-00 < Y2 < ∞ 2.

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