If X1,...Xn are normally distributed random variables. is the sample mean an unbiased estimator for the...

If X1,...Xn are normally distributed random variables. is the sample mean an unbiased estimator for the median? can you explain why or why not?

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orchestra answered 1 year ago

2. Let X1, X2, . . . , Xn be independent, uniformly distributed random variables on...

2. Let X1, X2, . . . , Xn be independent, uniformly distributed random variables on the interval [0, θ]. (a) Find the pdf of X(j) , the j th order statistic. (b) Use the result from (a) to find E(X(j)). (c) Use the result from (b) to find E(X(j)−X(j−1)), the mean difference between two successive order statistics. (d) Suppose that n = 10, and X1, . . . , X10 represents the waiting times that the n = 10...

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

Let X1, … , Xn be independent where Xi is normally distributed with unknown mean µ...

Let X1, … , Xn be independent where Xi is normally distributed with unknown mean µ and unknown variance o2 > 0. Find the likelihood ratio test for testing that µ = 0 against −∞ < µ < ∞.

Let X1, … , Xn be independent where Xi is normally distributed with unknown mean µ...

Let X1, … , Xn be independent where Xi is normally distributed with unknown mean µ and unknown variance o2 > 0. Find the likelihood ratio test for testing that µ = 0 against −∞ < µ < ∞.

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.

Suppose we have a random sample of n observations {x1, x2, x3,…xn}. Consider the following estimator...

Suppose we have a random sample of n observations {x1, x2, x3,…xn}. Consider the following estimator of µx, the population mean. Z = 12x1 + 14x2 + 18x3 +…+ 12n-1xn−1 + 12nxn Verify that for a finite sample size, Z is a biased estimator. Recall that Bias(Z) = E(Z) − µx. Write down a formula for Bias(Z) as a function of n and µx. Is Z asymptotically unbiased? Explain. Use the fact that for 0 < r < 1, limn→∞i=1nri...

Let X1,...,Xn be i.i.d. random variables with mean 0 and variance 2 > 0. In class...

Let X1,...,Xn be i.i.d. random variables with mean 0 and variance 2 > 0. In class we have shown a central limit theorem, ¯ Xn /pn )N(0,1), as n!1 , (1) with the assumption E(X1) = 0. Using (1), we now prove the theorem for a more general E(X1)=µ6=0 case. Now suppose X1,...,Xn are i.i.d. random variables with mean µ6= 0 and variance 2. (a) Show that for dummy random variables Yi = Xi µ, E(Yi) = 0 and V...

Let X1,…, Xn be a sample of iid N(0, ?) random variables with Θ=(0, ∞). Determine...

Let X1,…, Xn be a sample of iid N(0, ?) random variables with Θ=(0, ∞). Determine a) the MLE ? of ?. b) E(? ̂). c) the asymptotic variance of the MLE of ?. d) the MLE of SD(Xi ) = √ ?.

Let X1,…, Xn be a sample of iid random variables with pdf f (x; ?1, ?2)...

Let X1,…, Xn be a sample of iid random variables with pdf f (x; ?1, ?2) = ?1 e^(−?1(x−?2)) with S = [?2, ∞) and Θ = ℝ+ × ℝ. Determine a) L(?1, ?2). b) the MLE of ?⃗ = (?1, ?2). c) E(? ̂ 2).

Let X1,...,Xn be independent random variables,and let X=X1+...+Xn be their sum. 1. Suppose that each Xi...

Let X1,...,Xn be independent random variables,and let X=X1+...+Xn be their sum. 1. Suppose that each Xi is geometric with respective parameter pi. It is known that the mean of X is equal to μ, where μ > 0. Show that the variance of X is minimized if the pi's are all equal to n/μ. 2. Suppose that each Xi is Bernoulli with respective parameter pi. It is known that the mean of X is equal to μ, where μ >...

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If X1,...Xn are normally distributed random variables. is the sample mean an unbiased estimator for the...

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