[8] 3. For an MA (1) process where {%ı} is a scq ucoce of i.i.d. raodom variables from WN((),()")...
4. (3 points) Let X,.., X be an i.i.d. Bernoulli random variables with parameter p. Is it reasonable to use the exponential distribution to describe the prior distribution of p? Answer 'yes' or 'no ad exain
4. (3 points) Let X,.., X be an i.i.d. Bernoulli random variables with parameter p. Is it reasonable to use the exponential distribution to describe the prior distribution of p? Answer 'yes' or 'no ad exain
Problem 1.29. Prove the central limit theorem for a sequence of i.i.d. Bernoulli(p) random variables, where p e (0,1). Hint: Compute the moment generating function of the object you want the limit of and use Taylor's expansion to show that it converges to the moment generating function of a standard normal. (In fact, the same proof, but without the computation being so explicit, works for a general distribution, as long as the secono moment is finite. And then pushing the...
(a) Suppose that i, X2,... , In is an i.i.d. sample from Exp(1). Show that, for a standard normal random variable Z b) Show Г(n) by differencing both sides of the approximation in part a. Then set a -0 to get Stirling's Formula. 5. Suppose that Y is an id sample from Negative Binomial (n,p). Give a normal approximation of Yn use CLT, when n is large. 6. (Mandatory for Graduate Student. Extra credit for undergrad.) Let Ai, converges to...
P8.4 [Based on exercise 8.2 from Childers, 2nd ed.] Let Wn be an IID sequence of zero-mean Gaussian random variables with variance σ류. Define a discrete-time random process Xn-p Xn-1 + wn, n-1, 2, , where Xo-W) and p is a constant. (a) Find the mean function Hx(n). (b) Find the auto-correlation function Rx(n1,n2).
2. FoRECASTING wITH MA PROCESSES. (i) How to check the invertibility of an MA(1) model? (ii) Suppose we use an MA model to model the process represented in Figure 2. Write down the model and find the estimates of the coefficients. (15 marks) 200 225 250 275 300 325 350 375 400 Sample: 1 2000 Included observations: 1999 Autocorrelation Partial Correlation AC PAC 10.498 0.498 2-0.042-0.386 -0.081 0.220 40.042 -0.183 5 0.003 0.157 6 0.013-0.127 7 0.019 0.131 8 0.013-0.112...
Consider the regression model
where the εi are i.i.d. N(0,σ2) random variables, for i = 1, 2, . .
. , n.
(a) (4 points) Show βˆ is normally distributed with mean β and
variance σ2 . 1 1SXX
Question 6 Consider the regression model y = Bo + B12 + 8 where the €, are i.i.d. N(0,0%) random variables, for i = 1,2, ..., n. (a) (4 points) Show B1 is normally distributed with mean B1 and variances
2. Suppose Xi,X2,..., Xn are i.i.d. random variables such that a e [0, 1] and has the following density function: r (2a) (1a-1 where ? > 0 is the parameter for the distribution. It is known that E(X) = 2 Compute the method of moments estimator for a
Central Limit Theorem: let x1,x2,...,xn be I.I.D. random variables with E(xi)= U Var(xi)= (sigma)^2 defind Z= x1+x2+...+xn the distribution of Z converges to a gaussian distribution P(Z<=z)=1-Q((z-Uz)/(sigma)^2) Use MATLAB to prove the central limit theorem. To achieve this, you will need to generate N random variables (I.I.D. with the distribution of your choice) and show that the distribution of the sum approaches a Guassian distribution. Plot the distribution and matlab code. Hint: you may find the hist() function helpful
Q2 Suppose X1, X2, ..., Xn are i.i.d. Bernoulli random variables with probability of success p. It is known that p = ΣΧ; is an unbiased estimator for p. n 1. Find E(@2) and show that p2 is a biased estimator for p. (Hint: make use of the distribution of X, and the fact that Var(Y) = E(Y2) – E(Y)2) 2. Suggest an unbiased estimator for p2. (Hint: use the fact that the sample variance is unbiased for variance.) Xi+2...
3. Let ? and ? be i.i.d. exponential (1) random variables. Find the moment generating function of ?−?.