Suppose we collect n samplesX1,X2, . . . ,Xn from a uniform distribution U[θ,1]. We know the maximum likelihood estimator of θ is θ = min(Xi). (1) What is the CDF of θ? (2) What is the pdf of θ? (3) Find the bias of θ. (4) Find the variance of θ. (5) Find the MSE of θ.

Suppose you have a joint distribution of x and y where • x has population mean ug and population variance o? • y has population mean ly and population variance o, and the population covariance between them is Ery. The population Pearson correlation is then given by Ery OxJy We collect n pairs of data, (X;, Yi), i = 1,.. , n. Each pair is an independent draw from this distribution. (a) Suppose we estimate our population covariance with n 1 (x; – Ha)(yi – µy). i=1 Is this an unbiased estimator of the population covariance? Show why or why not. (b) Suppose we estimate our correlation with (xi – Ha)(Yi – Hy) - n i=1 Ox0y Is this an unbiased estimator of the population correlation? Show why or why not. =WI

Varza) (16) Let X =U(3) find x) and distribution function.

Let X - Poisson(A) and Y = VX. Use the delta method to approximate the mean and variance of Y,

N(0, o?) is For the simple linear model Y = a + BX + €, where the error variable e ~ independent of X, use the law of total variance to show that Var(Y) = 3² Var(X)+o².

Yearly returns on two assets (in percent) are denoted by rx and ry for asset X and Y respectively. Their joint distribution is multivariate normal with with mean (6, 2) and a covariance matrix 0.4 -0.2 -0.2 0.3 • Find the best linear approximation of the relationship between returns for a given return of asset X.

Q. Let the joint pmf of x and y be defined by f (x,y) = (x+y)/32  where x = 1,2 and y = 1,2,3,4