Let X1...Xn be independent, identically distributed random sample from a poisson distribution with mean theta.
a. Find the meximum liklihood estimator of theta, thetahat
b. find the large sample distribution for (sqrt(n))*(thetahat-theta)
c. Construct a large sample confidence interval for P(X=k; theta)
Let X1...Xn be independent, identically distributed random sample from a poisson distribution with mean theta. a....
Let X1,, Xn be independent and identically distributed random variables with unknown mean μ and unknown variance σ2. It is given that the sample variance is an unbiased estimator of ơ2 Suggest why the estimator Xf -S2 might be proposed for estimating 2, justify your answer
Let X1,X2,...,Xn be an independent and identically distributed (i.i.d.) random sample of Beta distribution with parameters α = 2 and β = 1, i.e., with probability density function fX(x) = 2x for x ∈ (0,1). Find the probability density function of the first and last order statistics Y1 and Yn.
2. Suppose that {X1, ..., Xn} are independent and identically distributed random variables from a distribution with p.d.f. See-ox if x > 0 f(x) = 10 if x = 0 Let Y = min <i<n X;. Find the p.d.f. of Y.
Let X1, ..., Xn denote an independent random sample from a population with a Poisson distribution with mean . Derive the most powerful test for testing Ho : 1= 2 versus Ha: 1= 1/2.
suppose X1 -> Xn is a random sample from a uniform distribution on the interval [0,theta]. let X1 = min {X1,X2,...Xn} and let Yn= nX1. show that Yn converges in distribution to an exponential random variable with mean theta.
Let X1,X2,...,Xn denote independent and identically distributed random variables with mean µ and variance 2. State whether each of the following statements are true or false, fully justifying your answer. (a) T =(n/n-1)X is a consistent estimator of µ. (b) T = is a consistent estimator of µ (assuming n7). (c) T = is an unbiased estimator of µ. (d) T = X1X2 is an unbiased estimator of µ^2. We were unable to transcribe this imageWe were unable to transcribe...
Question 6 Let X1, . . . , Xn denote a sequence of independent and identically distributed i.id. N(14x, σ2) random variables, and let Yı, . . . , Yrn denote an independent sequence of iid. Nụy, σ2) ran- dom variables. il Λί and Y is an unbiased estimator of μ for any value of λ in the unit interval, i.e. 0 < λ < 1. 2. Verify that the variance of this estimator is minimised when and determine the...
13. Let X1, X2, ...,Xy be a sequence of independent and identically distributed discrete random variables, each with probability mass function P(X = k)=,, for k = 0,1,2,3,.... emak (a) Find the expected value and the variance of the sample mean as = N&i=1X,. (b) Find the probability mass function of X. (c) Find an approximate pdf of X when N is very large (N −0).
1. (40) Suppose that X1, X2, Xn forms an independent and identically distributed sample from a normal distribution with mean μ and variance σ2, both unknown: 2nơ2 (a) Derive the sample variance, S2, for this random sample. (b) Derive the maximum likelihood estimator (MLE) of μ and σ2 denoted μ and σ2, respectively. (c) Find the MLE of μ3 (d) Derive the method of moment estimator of μ and σ2, denoted μΜΟΜΕ and σ2MOME, respectively (e) Show that μ and...
Let X1, X2, ...,Xn be a random sample of size n from a Poisson distribution with mean 2. Consider a1 = *1782 and în = X. Find RE(21, 22) for n = 25 and interpret the meaning of the RE in the context of this question.