1- X[n] is a Bernoulli random process that takes on values -1 or 1, each with probability of p = ½. Is Y[n] an IID random process in each case? a) Y[n] = (-1)"X[n] b) Y[n] = cos √3 (¹7+X[n] (Hint: cos ( (²) =/; sin (7) = 1) 2
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- A time series with a periodic component can be constructed fromxt = U1 sin(2πω0t) + U2 cos(2πω0t),where U1 and U2 are independent random variables with zero means andE(U21 ) = E(U22 ) = σ2. The constant ω0 determines the period or time ittakes the process to make one complete cycle. Show that this series is weaklystationary with autocovariance functionγ(h) = σ2 cos(2πω0hConsider a series xt generated by the moving average process as: xt = µ + εt + θ1εt−1, where εt are independently identically distributed random variables with E(εt) = 0, and V ar(εt) = σ2. Calculate the unconditional mean and the unconditional variance of xt. What is meant by saying that a process like xt is invertible? What condition would assure that xt is invertible? If θ = 0.75, does xt satisfy the invertibility condition? What shapes of the ACF and PACF functions do you expect for xt? Derive the first 4 autocorrelations for this process (τ1 up to τ4). Carefully write the equations for the 1, 2, 3 and 4 step ahead forecasts for xt.Suppose X1, X2, ... , Xn is a random sample and Xi = {1, with probability p 0, with probability 1-p} for every i = 1, 2, ... , n. Find the Moment Generating Function of ∑i=1n Xi . What is the distribution of ∑i=1n Xi ?
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- A single observation of a random variable having a hypergeometric distribution with N = 7 and n = 2 is used to test the null hypothesis k = 2 against the alternative hypothesis k = 4. If the null hypothesis is rejected if and only if the value of the random variable is 2, find the power of the test.Let Xi be iid Exp(λ) random variables.1. Let Mn = max1≤i≤n Xi. Compute the CDF FMn (x).2. Using the identity P(X > x) = exp(−λx), show that if an = log n, and Yn = Mn − an, thenFYn(x) → e^(-e^-x).3. * What does this say, heuristically, about the order of magnitude of Yn?A poisson random variables has f(x,3)= 3x e-3÷x! ,x= 0,1.......,∞. find the probabilities for X=0 1 2 3 4 and also find mean and variance from f(x,3).?