1. Let X1, X2,.., Xn be a random sample from the distribution with the probability density function f(x; 0) = (0 + 1)(1 − x) 0, 0 < x < 1, 0 > -1. a) Find a unidimensional sufficient statistic Y = u(X1, X2, ..., Xn) for 6. b) Determine the Fisher information I(0).

Let X and Y are two random variables with p.d.f as F(x,y) = 3x^2y+3xy^2,0 < x< 1,0 < y < 1. (i) Find conditional density of X given Y. (ii) Find the mean, mode and standard deviation of r.v Y. (iii) Correlation between X and Y. (iv) Check that X and Y are independent or not?

Let X1, . X,, be n independent random variables with PDF laje- if x > 0, 10 fx, (x) = i = 1,2, ...,n. if x, 0 V i. Find the distribution of Y = min{X1, X2,..., X,) and Z = max{X1, X2,..., X„}.

Let X, Y be independent random variables with exponential distribution of parameter θ > 0. Are the random variables Z = X + Y and W = X / (X+Y) independent?

Let X be a continuous random variable with PDF 3 x > 1 x4 fx(x) = otherwise Find the mean and variance of x.

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).

b) Let Y,,Y2, .. , Yn denote a random sample from N(0,0) distribution with probability density function: f(y;8) = e V2n0 i) Show that f(y; 0) belongs to the 1-parameter exponential family. ii) What is the complete sufficient statistic for 0? Justify your answer. iii) Show whether or not, the maximum likelihood estimator is an unbiased estimator of 0. iv) Does the estimator attains the minimum variance unbiased estimator of 0.

Let f(x, y) = x + y for 0 < x < 1 and 0 < y < 1 The Conditional Variance of Y when X = ; is

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, X₂, ..., Xn constitute a random sample from the probability density function given by f(x,0)= 2 02 -(1-x), 0 < y < 0 (₁ 0, elsewher construct a 90% confidence interval for