Suppose X1, X2, . . ., Xn are iid Poisson random variables with parameter λ. (a)...

Suppose X1, X2, . . ., Xn are iid Poisson random variables with parameter λ.

(a) Find the MVUE for λ.

(b) Find the MVUE for λ 2

Expert Solution

Note-if there is any understanding problem regarding this please feel free to ask via comment box..thank you

orchestra answered 2 years ago

Let X1, X2, . . . , Xn be iid Poisson random variables with unknown mean...

Let X1, X2, . . . , Xn be iid Poisson random variables with unknown mean µ 1. Find the maximum likelihood estimator of µ 2.Determine whether the maximum likelihood estimator is unbiased for µ

1) Suppose X1,X2,...,Xn are iid Gamma(α,λ) random variables. a) Express the ﬁrst and third moments (µ1...

1) Suppose X1,X2,...,Xn are iid Gamma(α,λ) random variables. a) Express the ﬁrst and third moments (µ1 = E(X) and µ3 = E(X3)) as functions of λ and α. b) Solve this system of equations to ﬁnd estimators of ˆ λ and ˆ α as functions of the empirical moments. Note that the estimates must be positive. 2) Suppose that X1,X2,...,Xn are a iid Beta(a, a) random variables. That is, a beta distribution with the restriction that b = a. Using...

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...

Suppose X1, X2, . . . are a sequence of iid uniform random variables with probability...

Suppose X1, X2, . . . are a sequence of iid uniform random variables with probability density function f(x) = 1 for 0 < x < 1, or 0, otherwise. Let Y be a continuous random variable with probability density function g(y) = 3y2 for 0 < y < 1, or 0, otherwise. We further assume that Y and X1, X2, . . . are independent. Let T = min{n ≥ 1 : Xn < Y }. That is, T...

Let X1,...,Xn be exponentially distributed independent random variables with parameter λ. (a) Find the pdf of...

Let X1,...,Xn be exponentially distributed independent random variables with parameter λ. (a) Find the pdf of Yn= max{X1,...,Xn}. (b) Find E[Yn]. (c) Find the median of Yn. (d) What is the mean for n= 1, n= 2, n= 3? What happens as n→∞? Explain why.

Let X1, X2,..., Xnbe independent and identically distributed exponential random variables with parameter λ . a)...

Let X1, X2,..., Xnbe independent and identically distributed exponential random variables with parameter λ . a) Compute P{max(X1, X2,..., Xn) ≤ x} and find the pdf of Y = max(X1, X2,..., Xn). b) Compute P{min(X1, X2,..., Xn) ≤ x} and find the pdf of Z = min(X1, X2,..., Xn). c) Compute E(Y) and E(Z).

Let X1,X2,...,Xn be i.i.d. Gamma random variables with parameters α and λ. The likelihood function is...

Let X1,X2,...,Xn be i.i.d. Gamma random variables with parameters α and λ. The likelihood function is diﬃcult to diﬀerentiate because of the gamma function. Rather than ﬁnding the maximum likelihood estimators, what are the method of moments estimators of both parameters α and λ?

Suppose X1, X2, ..., Xn iid∼ Multinomial (p). a) What is the MLE for the situation...

Suppose X1, X2, ..., Xn iid∼ Multinomial (p). a) What is the MLE for the situation where p = (p1, p2, p3, p4), such that p1 = p2 and p3 = p4? b) What can you interpret from these estimates in terms of the number of observations of type k (k = 1, 2, 3, 4) compared the total number of trials n?

Let X1, X2, . . . be iid random variables following a uniform distribution on the...

Let X1, X2, . . . be iid random variables following a uniform distribution on the interval [0, θ]. Show that max(X1, . . . , Xn) → θ in probability as n → ∞

) Let X1, . . . , Xn be iid from the distribution with parameter η...

) Let X1, . . . , Xn be iid from the distribution with parameter η and probability density function: f(x; η) = e ^(−x+η) , x > η, and zero otherwise. 1. Find the MLE of η. 2. Show that X_1:n is sufficient and complete for η. 3. Find the UMVUE of η.

Question