taken from the uniform distribution on the interval [0,0] and that θ is unknown. How large...
3. Let X X be a random sample from Uniform[0, where > 0 is unknown. (a) Show that = max{X,X is the MLE of 0. (b) Let the CDF of @ be F(-). Find F(t) for any t e R (c) Find the pdf of 0 Hint: Find the distribution function of Z maxX1,X. The first feu steps will be as follous: F2(2) P(Z) P (maxX, x) ) = P (XS2, X X,) Nert use the fact that Xis are...
5. Let X1,.. ., Xn be a random sample from Uniform(0,0) with an unknown endpoint θ > 0, we want to estimate the parameter θ (a) Find the method of moments estimator (MME) of θ. (b) Find the MLE θ of θ (c) (R) Set the sample size as 25, do a simulation in R to compare these two esti- mators in terms of their bias and variance. Include a side-by-side boxplot that compares their sampling distributions
A random variable, X, has uniform distribution on the interval [0,θ] where θ is unknown. A hypothesis test is as follows: H0: θ = 2 H1: θ ≠ 2 It has been decided to reject H0 if the observed value of x is x ≤ 0.1 or x ≥ 1.9. Part a: Find the probability of committing a Type I error. Part b: Suppose the true value of θ is 3. Find the probability of committing a Type II error....
Suppose a simple random sample (Xi i.i.d. observations) was taken from a uniform[0,θ] distribution with an unknown parameter θ Sample observations are below: 1.984, 3.551, 2.367, 3.201, 3.690, 3.256, 2.949, 1.872, 3.357, 2.026, 2.358, 3.337, 1.582, 0.902, 0.718, 1.850, 2.026, 1.326, 3.491 Let τ=P(X≤0.50). MLE of τ
3. Suppose that Xi,.... Xn is a random sample from a uniform distribution over [0,0) That is, 0 elsewhere Also suppose that the prior distribution of θ is a Pareto distribution with density 0 elsewhere where θ0 > 0 and α > 1. (a) Determine (b) Show , θ > max(T1 , . . . ,Zn,%) and hence deduce the posterior density of θ given x, . . . ,Zn is (c) Compute the mean of the posterior distribution and...
Problem 2.2 Suppose observations will be taken from a uniform distribution on the interval [e-0.5, e+ 0.5] where the value of θ is unknown, and the prior distribution of θ is a uniform distribution on the interval [10,20] a) If a single observation is selected and yields x-11, determine the posterior distribution of. ξ(9111) b) Ifn - 4 bservations are selected and yield data: 11, 11.5, 11.7, 10.9 then determine the posterior distribution of 0. ξ(θ| 11,1 15,1 1.7, 10.9)...
Let X,X,, X, be a random sample of size 3 from a uniform distribution having pdf /(x:0) = θ,0 < x < 0,0 < θ, and let):く,), be the corresponding order statistics. a. Show that 2Y, is an unbiased estimator of 0 and find its variance. b. Y is a sufficient statistic for 8. Determine the mean and variance of Y c. Determine the joint pdf of Y, and Y,, and use it to find the conditional expectation Find the...
Let X1,X2Xn be a random sample from a uniform distribution on the interval (0,0) (a) Show that the density function of Xcp-minXXXn) is given by n-1 72 0 otherwise (b) Use (a) to calculate E[Xcu]. Calculate the bias, B(6). Find a function of Xo) that is an unbiased estimator of 0
4. Let X,x, X, be a random sample from a uniform distribution on the interval (0,0) (a) Show that the density function of XnX,X2 Xn is given by 0 otherwise (b) Use (a) to calculate E[X)). Caleulate the bias, B). Find a function of X) that is an unbiased estimator of .
Let Xi., Xn be a random sample from the distribution with density f(r, θ)-303/2.4 for x > θ and 0 otherwise. Determine the MLE of θ and derive 90% central CI interval for θ. If possible find an exact CI. Otherwise determine an approximate CI. Explain your choice Let Xi., Xn be a random sample from the distribution with density f(r, θ)-303/2.4 for x > θ and 0 otherwise. Determine the MLE of θ and derive 90% central CI interval...