2. Use the method of maximum likelihood to estimate 0 in the pdf V, y0. fr (y; 0) 2/y Evaluate 0e for the following random sample of size 4: yi 6, y2 = 8, y3 = 2.4, y4 = 5.9.
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- Suppose X, Y, Z are iid observations from a Poisson distribution with parameter λ, which is unknown. Consider the 3 estimators T1 = X + Y − Z, T2 = 2X + Y + Z 4 , T3 = 3X + Y + Z 5 . (a) Which among the above estimators are unbiased? (b) Among the class of unbiased estimators, which has the minimum variance?Consider a random sample X_1, X_2, ... , X_n from the pdf f(x; θ) = (θ+ 1)x^θ 0≤x≤1 where θ > −1 Use the Maximum Likelihood Method to obtain an estimator of θ and then compute the estimatefor the following random sample of four observations: 0.42, 0.37, 0.64, and 0.55.Let X1, ..., Xn be a sample from an exponential population with parameter λ.(a) Find the maximum likelihood estimator for λ. (b) Is the estimator unbiased?(c) Is the estimator consistent?
- Let X1, ..., Xn be a sample from a geometric random variable with parameter p.(a) Find the maximum likelihood estimator for p.(b) Is the estimator unbiased?(c) Is the estimator consistent?If X1,X2,...,Xn constitute a random sample of size n from a gamma population with α =2, use the method of maximum likelihood to find a formula for estimating β.If X is a uniformly distributed random varibale with a=9 and b=16, then Calculate the variance of X? Round to three decimal places
- If X1 and X2 constitute a random sample of size n = 2from a Poisson population, show that the mean of thesample is a sufficient estimator of the parameter λ.A snack food manufacturer estimates that the variance of the number of grams of carbohydrates in servings of its tortilla chips is 1.34. A dietician is asked to test this claim and finds that a random sample of 16 servings has a variance of 1.22. At α=0.05, is there enough evidence to reject the manufacturer's claim? Assume the population is normally distributed. Complete parts (a) through (e) below.Assume X_1, X_2, .....X_n are random samples from X~Exponential(θ).1) Find the MLE estimator of θ .
- Let the following simple random sample following:1. Binomial pmf. (11, ¾);2. Uniform pmf;3. Uniform pdf (0, a);4. Exponential pdf with (µ) .Find the corresponding pmf/pdf of Y1 , Y4 , Y7 and F(Yi) whereIf y1, y2,..., ym be a random sample taken from a normal distribution with parameters x and n× n, then the likelihood equation isSuppose X1, . . . , Xn be a random sample from the Beta(θ, 1) distribution. Find the P-value for the LRT test of the hypotheses H0 : θ ≥ 1 vs H1 : θ < 1