X is a random variable following Poisson distribution. X1 is an observation (random sample point) of...

X is a random variable following Poisson distribution. X1 is an observation (random sample point) of X.

(1.1) Please find probability distribution of X and X1. Make sure to define related parameter properly.

(1.2) Please give the probability distribution of a random sample with sample size of n that consists of X1, X2, ..., Xn as its observations.

(1.3) Please give an approximate distribution of the sample mean in question 1.2(say, called Y) when sample size is 100 with detailed explanation.

Expert Solution

thanks

orchestra answered 2 years ago

Suppose X1, X2, ..., Xn is a random sample from a Poisson distribution with unknown parameter...

Suppose X1, X2, ..., Xn is a random sample from a Poisson distribution with unknown parameter µ. a. What is the mean and variance of this distribution? b. Is X1 + 2X6 − X8 an estimator of µ? Is it a good estimator? Why or why not? c. Find the moment estimator and MLE of µ. d. Show the estimators in (c) are unbiased. e. Find the MSE of the estimators in (c). Given the frequency table below: X 0...

1. If the random variable x has a Poisson Distribution with mean μ = 53.4, find...

1. If the random variable x has a Poisson Distribution with mean μ = 53.4, find the maximum usual value for x. Round your answer to two decimal places. 2. In one town, the number of burglaries in a week has a Poisson distribution with mean μ = 7.2. Let variable x denote the number of burglaries in this town in a randomly selected month. Find the smallest usual value for x. Round your answer to three decimal places. (HINT:...

Given that xx is a random variable having a Poisson distribution, compute the following: (a) P(x=1)P(x=1) when...

Given that xx is a random variable having a Poisson distribution, compute the following: (a) P(x=1)P(x=1) when μ=4.5μ=4.5 P(x)=P(x)= (b) P(x≤8)P(x≤8)when μ=0.5μ=0.5 P(x)=P(x)= (c) P(x>7)P(x>7) when μ=4μ=4 P(x)=P(x)= (d) P(x<1)P(x<1) when μ=1μ=1 P(x)=P(x)=

A single observation of a random variable (that is, a sample of size n = 1)...

A single observation of a random variable (that is, a sample of size n = 1) having a geometric distribution is used to test the null hypothesis θ = θ0 against the alternative hypothesis θ = θ1 for θ1 < θ0. The null hypothesis is rejected if the observed value of the random variable is greater than or equal to some positive integer k. Find expressions for the probabilities of type I and type II errors.

The demand for a replacement part is a random variable having a Poisson probability distribution with...

The demand for a replacement part is a random variable having a Poisson probability distribution with a mean of 4.7. If there are 6 replacement parts in stock, what is the probability of a stock-out (i.e., demand exceeding quantity on hand)?

2. Let X1, . . . , Xn be a random sample from the distribution with...

2. Let X1, . . . , Xn be a random sample from the distribution with pdf given by fX(x;β) = β 1(x ≥ 1). xβ+1 (a) Show that T = ni=1 log Xi is a sufficient statistic for β. Hint: Use n1n1n=exp log=exp −logxi .i=1 xi i=1 xi i=1 (b) Find the pdf of Y = logX, where X ∼ fX(x;β). (c) Find the distribution of T . Hint: Identify the distribution of Y and use mgfs. (d) Find...

Let X1, . . . , Xn be a random sample from a uniform distribution on...

Let X1, . . . , Xn be a random sample from a uniform distribution on the interval [a, b] (i) Find the moments estimators of a and b. (ii) Find the MLEs of a and b.

1)What is a sampling distribution for the sample mean, x-bar? Explain. Is x-bar a random variable?...

1)What is a sampling distribution for the sample mean, x-bar? Explain. Is x-bar a random variable? Why do you think so? 2) We learned standard error of the mean ( read sigma xbar in denotation) or standard deviation of sample mean. When the size(n) of the sample collected from a infinite population is greater than 1, why is the standard error of the mean smaller than the standard deviation of X, population ? Explain conceptually or/and algebraically. Think of formula for...

Suppose that random variable X 0 = (X1, X2) is such that E[X 0 ] =...

Suppose that random variable X 0 = (X1, X2) is such that E[X 0 ] = (µ1, µ2) and var[X] = σ11 σ12 σ12 σ22 . (a matrix) (i) Let Y = a + bX1 + cX2. Obtain an expression for the mean and variance of Y . (ii) Let Y = a + BX where a' = (a1, a2) B = b11 b12 0 b22 (a matrix). Obtain an expression for the mean and variance of Y . (ii)...

Let X be a random variable with the following probability distribution: Value x of X P(X=x) ...

Let X be a random variable with the following probability distribution: Value x of X P(X=x) 20 0.35 30 0.10 40 0.25 50 0.30 Find the expectation E (X) and variance Var (X) of X. (If necessary, consult a list of formulas.) E (x) = ? Var (X) = ?

Question