Suppose X has an exponential distribution with parameter A = 1 and Y|X = # has a Poisson distributionwith parameter z.a) Generate at least 1000 random samples from the marginal distribution of X and make a probabilityhistogram.b) Generate at least 1000 random samples from the conditional distribution of Y|X = 1.5 and make aprobability histogram.c) Generate at least 1000 random samples from the marginal distribution of Y and make a probabilityhistogram.d) Generate at least 1000 random samples from the conditional distribution of X|Y = 2 and make aprobability histogram.

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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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Person P is walking with his dog every day. He decides to model the time intervals between dog encounters (i.e., how long does it take between two encounters) with the independent random variables Y1, . . ., Yn. Number of dogs encounters per day, the number of customers entering a store per hour, the number of bacteria per liter of water, etc. are classically modelled using the Poisson distribution. With an exponential distribution, on the other hand, is used for modelling intervals of between events and how long a light bulb lasts, for example. After learning this, person P decides to model the observations in such a way that the corresponding random variables Y1,..., Yn for a random sample, that is, independent observation, of the exponential distribution Exp(1/µ), µ > 0. The parameter μ > 0 is used as the statistical model parameter 1 which is the expected value of each observation random variable Yi. Derive (a) the joint density function f(y;µ) specifying the model. (b) the…
Consider an auto insurance portfolio where the number of accidents follows a Poisson distribution with parameter lamda = 1000. Suppose the damage sizes for separate accidents are i.i.d. (independent identically distributed) r.v.'s having an exponential distribution with a mean of $2500. Each policy involves a deductible of $500. Let N1 be the number of accidents that result in claims, and N2 be the number of accidents that do not result in claims. Answer the following questions 1-5. 1. Are N1, N2 dependent? 2. What is the name of the distributions of N1, N2? Compound Poisson or Marked Poisson? 3. What is the mean and variance of N1? 4. What is the mean and variance of N2? 5. Which of the number is close to the probability that N1 will not be larger than 790? (Hint: to what distribution is the Poisson distribution with a parameter of lamda close for "large" lamda's?
Let X1 and X2 are each independent Bernoulli random variable with probability of success p=0.2 and Y= X1 -3 X22. Compute the expected value of Y. Let X1 and X2 are each independent Exponential random variable with mean 5 and Y= X1 -3 X2 then Compute the value of variance of Y. Let X has Poisson distribution with mean 10. Compute correlation between X and -
3. Let Y₁ 5)
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…
Consider an auto insurance portfolio where the number of accidents follows a Poisson distribution with parameter λ= 1000. Suppose the damage sizes for separate accidents are i.i.d. (independent identically distributed) r.v.'s having an exponential distribution with a mean of $2500. Each policy involves a deductible of $500. Let N₁ be the number of accidents that result in claims, and N₂ be the number of accidents that do not result in claims. Answer the following questions 1-5. Q1 Are N₁, N₂ dependent? Q2 Depends on a situation, No Yes What is the name of the distributions of N₁, №₂? O Marked Poisson Gamma Exponential Compound Poisson
The number of customers arriving per hour at a certain automobile service facility is assumed to follow a Poisson distribution with mean λ = 7. (a) Compute the probability that more than 12 customers will arrive in a 2-hour period. (b) What is the mean number of arrivals during a 2-hour period.
Suppose X is a random variable of uniform distribution between 1 and 7. Find E(X)
Let X1, X2,..., Xn be a random sample of size n > 3 from a normal distribution with unknown mean μ and known variance equal to 2. Show that is an unbiased estimator of μ and compute its variance. p
Person P is walking with his dog every day. He decides to model the time intervals between dog encounters (i.e., how long does it take between two encounters) with the independent random variables Y₁, . . ., Yn . Number of dogs encounters per day, the number of customers entering a store per hour, the number of bacteria per liter of water, etc. are classically modelled using the Poisson distribution. With an exponential distribution, on the other hand, is used for modelling intervals of between events and how long a light bulb lasts, for example. After learning this, person P decides to model the observations in such a way that the corresponding random variables Y1,..., Yn for a random sample, that is, independent observation, of the exponential distribution Exp(1/µ), µ > 0. The parameter µ > 0 is used as the statistical model parameter 1 which is the expected value of each observation random variable Yi. (a) Form the maximum likelihood estimate μ^(y) in the model for the parameter μ…

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