

01) Let X1, X2, ...., Xm be m independent growth observations on the first type with...
Question 1 Let Y = 1 +2X + u where X = Zn4Z2, u = Z1-22, Z1 and Z2 are independent standard normals. We have iid observations (X,, Y1 from this model. (a) Suppose we run the following regression to obtain the OLS estimator β0Ls. What would you expect the value of β0LS to be when no Please give a numerical answer (b) Suppose we run the following regression to obtain the OLS estimator γ0Ls. What would you expect the...
Suppose X1, .. ,XM are independent, identically distributed random variables with mean a and variance b2. Let aM ≡ (1/M)Σi=1M aM and bM2≡ (1/(M-1)) Σi=1M (Xi-aM)2. a) Show that aM is an unbiased estimator of E[X]: that is, E[aM] = a. b) Assume that the identity E[ Σi=1M (Xi-aM)2 ] = (M-1) b2 is correct. Show that bM2 is an unbiased estimator of var(X): that is, E[bM2] = b2
3. Let Ya» . . . , Yn be independent normally distributed random variables with E(X) Gai and V(X)-1. Recall that the normal density with mean μ and variance σ given by TO 202 (a) Find the maximum likelihood estimator β of β (b) Show that ß is unbiased. (c) Determine the distribution of β (d) Recall that the likelihood ratio test of Ho : θ 02] L1] L2] θ° is to θ0 against H1: θ reject Ho if L(e)...
e (4 marks) Let m be an integer with the property that m 2 2. Consider that X1, X2,.. ., Xm are independent Binomial(n,p) random variables, where n is known and p is unknown. Note that p E (0,1). Write down the expression of the likelihood function We assume that min(x1, . . . ,xm) 〈 n and max(x1, . . . ,xm) 〉 0 5 marks) Find , and give all possible solutions to the equation dL dL -...
1. Suppose that E(X) E(Y) E(Z) 2 Y and Z are independent, Cov(X, Y) V(X) V(Z) 4, V(Y) = 3 Let U X 3Y +Z and W = 2X + Y + Z 1, and Cov(X, Z) = -1 Compute E(U) and V (U) b. Compute Cov(U, W). а.
Question 3 15 marks] Let X1,..,X be independent identically distributed random variables with pdf common ) = { (#)%2-1/64 0 fx (a;e) 0 where 0 >0 is an unknown parameter X-1. Show that Y ~ T (}, ); (a) Let Y (b) Show that 1 T n =1 is an unbiased estimator of 0-1 ewhere / (0; X) is the log- likeliho od function; (c) Compute U - (d) What functions T (0) have unbiased estimators that attain the relevant...
Number 3.
UCD515 IUL NEL 1). JupUSC LIS 15 DALL -ste sis for V: {x1,x2,...,x. Let y = T(x) for i=k+1, k + 2,...,n. Show that {Yk+1, Yt+2, ..., yn) is a basis of image(T). 3. Prove or Disprove: There exists three distinct subspaces U, V and W of Rº such that R =U V and R3 = U W . (Recall, e denotes a direct sum)
(a) Let X, have a chi-squared distribution with parameter V, and let X, be independent of X, and have a chi-squared distribution with parameter vz. Show that X, + X, has a chi-squared distribution with parameter v, + V Let Y = X1 + Xy. Identify the correct expression for Fly). Fyly) = (f1 +49 (0) BM={{1*(**) 2*3)**- _jei OFW - 1 -{{49)..:@) ib) dx1 FY) = -xq12 dx + -*2/20 1²ax 2 1/2 O Fy(y) :{"(****): 19). x 22...
Let X1 d= R(0,1) and X2 d= Bernoulli(1/3) be two independent random variables, define Y := X1 + X2 and U := X1X2. (a) Find the state space of Y and derive the cdf FY and pdf fY of Y . (You may wish to use {X2 = i}, i = 0,1, as a partition and apply the total probability formula.) (b) Compute the mean and variance of Y in two different ways, one is through the pdf of Y...
2. Let X1,.n be a random sample from the density 0 otherwise Suppose n = 2m+ 1 for some integer m. Let Y be the sample median and Z = max(Xi) be the sample maximum (a) Apply the usual formula for the density of an order statistic to show the density of Y is (b) Note that a beta random variable X has density f(x) = TaT(可 with mean μ = α/(a + β) and variance σ2 = αβ/((a +s+...