be a random sample of size n and given a >( >0 with the probability density function of X is f(x;a,0) =xª where 0
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- If X is a continuous random variable find the CDF and density of the function y = x/3Let X,, X2,... ,X, be a random sample from a distribution with density function if x20 f(x; 0) elsewhere What is the maximum likelihood estimator of 0 ?Let X1, X2,...,X, be a random sample from a distribution with density function e if x > 0 f(x; 0) elsewhere What is the maximum likelihood estimator of 0 ?
- Let X1 and X2 be two continuous random variableshaving the joint probability density f(x1, x2) = 4x1x2 for 0 < x1 < 1, 0 < x2 < 10 elsewhereFind the joint probability density of Y1 = X21 and Y2 = X1X2.the popularity density function f(x) for a uniform random variable X defined over the interval [2,10] is?Let X1,., X, denote a random sample from the probability density function f(r) = 0xº-1, 0 0. Show that X is a consistent estimator of .
- Let Y1 < Y2 < Y3 be the order statistics of a random sample of size 3 froma distribution having the pdf f(x) = 2x, 0 < x < 1, zero elsewhere. Show thatZ1 = Y1/Y2, Z2 = Y2/Y3, and Z3 = Y3 are mutually independent.Let x1,x2,..x n be an jid sample with pdf f(x|0)=0X (® -1) OSxs1,0>0 Find the MLE of 0.Let X1,., X, denote a random sample of size n from the population with probability density function 16) = {** 381-4, r2 B 0, otherwise where 3 > 0 is the population parameter we want to estimate. Consider the estimator B = min(X1,.,X„). [The estimator is the 1st-order order statistic of the random sample.] (a) Derive the bias of the estimator 3. (b) Derive the mean squared error of 3, MSE(3).
- 2. Let X be a random variable with density function ) = √(2x² - + f(x) = Find the expected value of g(x) = 2X+1. 8x2 +5x + 6), 1 < x < 2, 0, elsewhereLet Xand Y be two continuous random variables with joint probability density [3x function given by: f(x.y)%D 0sysxsl elsewhere with E(X) = ECX)- EC) - EC*)= ;and E(XY) = 10 3 E(Y*) = - and E(XY) =; %3D Then the value of the variance of 2X+Y is: O 3/80 O 91/320 43/320 7/20The probability density of the random variable Z isgiven by f(z) = kze−z2for z > 00 for z F 0Find k and draw the graph of this probability density.