5. (10 marks) (a) Let E, E2} be mutually independent random variables. Show that the conditional...
Question 4 15 marks] The random variables X1, ... , Xn random variables with common pdf independent and identically distributed are 0 E fx (x;01) 0 independent of the random variables Y^,..., Y, which and are indepen are dent and identically distributed random variables with common pdf 0 fy (y; 02) 0 (a) Show that the MLE8 of 01 and 02 are 1 = X i=1 Y (b) Show that the MLE of 0 when 01 = 0, = 0...
Question 4 [15 marks] The random variables X1,... , Xn are independent and identically distributed with probability function Px (1 -px)1 1-2 -{ 0,1 fx (x) ; otherwise, 0 while the random variables Yı,...,Yn are independent and identically dis- tributed with probability function = { p¥ (1 - py) y 0,1,2 ; otherwise fy (y) 0 where px and py are between 0 and 1 (a) Show that the MLEs of px and py are Xi, n PY 2n (b)...
(10 marks) Let X1, X2,... be a sequence of independent and identically distributed random variables with mean EX1 = i and VarX1 = a2. Let Yı, Y2, ... be another sequence of independent and identically distributed random variables with mean EY = u and VarY1 a2 Define the random variable ( ΣxΣ) 1 Dn 2ng2 i= i=1 Prove that Dn converges in distribution to a standard normal distribution, i.e., prove that 1 P(Dn ) dt 2T as n >oo for...
Let exp(-т*) + vk Yk where dent M and V N(0, o2 are mutually indepen R, k = 1, (a) Construct the likelihood T(y|x) and the negative log-likelihood. (b) Compute the maximum likelihood estimate îML (c) Bonus question: How does the estimate change if E(k) t0? Let exp(-т*) + vk Yk where dent M and V N(0, o2 are mutually indepen R, k = 1, (a) Construct the likelihood T(y|x) and the negative log-likelihood. (b) Compute the maximum likelihood estimate...
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...
Let X1, X2, ..., Xn be independent Exp(2) distributed random vari- ables, and set Y1 = X(1), and Yk = X(k) – X(k-1), 2<k<n. Find the joint pdf of Yı,Y2, ...,Yn. Hint: Note that (Y1,Y2, ...,Yn) = g(X(1), X(2), ..., X(n)), where g is invertible and differentiable. Use the change of variable formula to derive the joint pdf of Y1, Y2, ...,Yn.
8. Let X1...., X, be i.i.d. ~E(1) random variables (i.e., they are independent and identically distributed, all with the exponential distribution of parameter 1 = 1). a) Compute the cdf of Yn = min(X1,...,xn). b) How do P({Y, St}) and P({X1 <t}) compare when n is large and t is such that t<? c) Compute the odf of Zn = max(X1...., X.). d) How do P({Zn2 t}) and P({X1 2 t}) compare when n is large and t is such...
4. (24 marks) Suppose that the random variables Yi,..., Yn satisfy Y-B BX,+ Ei, 1-1, , n, where βο and βι are parameters, X1, ,X, are con- stants, and e1,... ,en are independent and identically distributed ran- dom variables with Ei ~ N (0,02), where σ2 is a third unknown pa- rameter. This is the familiar form for a simple linear regression model, where the parameters A, β, and σ2 explain the relationship between a dependent (or response) variable Y...
Question 3 [17 marks] The random variable X is distributed exponentially with parameter A i.e. X~ Exp(A), so that its probability density function (pdf) of X is SO e /A fx(x) | 0, (2) (a) Let Y log(X. When A = 1, (i) Show that the pdf of Y is fr(y) = e (u+e-") (ii) Derive the moment generating function of Y, My(t), and give the values of t such that My(t) is well defined. (b) Suppose that Xi, i...
Let d ln(B)e where the unknown 3 E R, k = 1,..., M and ek are dent and identically distributed according to the (standard) Cauchy distribution, that is mutually indepen- 1 1 т (еx) k (a) Construct the likelihood T(d|B) and the negative log-likelihood (b) In this case, you cannot find an likelihood estimate BML But do this: derive the nonlinear equa- explicit form for the maximum tion of the form (- In т(d|B)) — F(83; d,, .., dм) dp...