(c). X, X,...,x, is a random sample from a population distributed as uniform(0,0). Let Y, =...
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 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...
18. Let X, X2, ..., Xv are independent and identically distributed standard uniform random variables. Find the following expectations: (a) E[max(X,,X2, .Xn,)] (b) E[min(X1,X2,..., Xn)]
.A Uniform random variable is defined on the interval (8,0+1). Let YY, be a random sample taken from this distribution and Y is an estimator of θ. (a) Compute the bias of Y (b) Find a function of that is an unbiased estimator of θ. (c) Find MSE(Y
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...
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
Let X 1 and X 2 be statistically independent and identically distributed uniform random variables on the interval [ 0 , 1 ) F X i ( x ) = { 0 x < 0 x 0 ≤ x < 1 1 x ≥ 1 Let Y = max ( X 1 , X 2 ) and Z = min ( X 1 , X 2 ) . Determine P(Y<=0.25), P(Z<=0.25), P(Y<=0.75), and P(Z<=0.75) Determine
+1,20] distribution, where , X.) 3. Let Xi,...,X be a random sample from Uniform θ > 1 is unknown. Let X(1)-min(X, , X.) and X() = max(X,, (a) Derive the edf of X(n) and then its pdf. b) Derive EoX(n) (c) Find a function g(X(n)) such that Eolg(X())-θ for all θ > 1. (d) Replace X(n) by Xu) in the above questions, parts (a) - (c), and answer those.
6. Let X1,..., Xn be a random sample from Uniform (0, 1). a) Find the exact distribution of U = – log(X(1)) where X(1) = min(X1, X2,..., Xn). b) Find the limiting distribution of n(1 – X(n)), where X(n) = max(X1, X2, ..., Xn).
3. (Bpoints) Let X, Y and Z be independent uniform random variables on the interval (0, 2), Let W min(X, y.z a) Find pdf of W Find E(1-11 b) 3. (Bpoints) Let X, Y and Z be independent uniform random variables on the interval (0, 2), Let W min(X, y.z a) Find pdf of W Find E(1-11 b)