i=1 Consider the random walk stochastic process Sn = So + Xi, where So = 0, and the independent identically distributed (iid) X; are the modified Bernoulli random variables taking values 2 and -1 with proba- bilities p and 1-p, respectively. a. Calculate E[Sn] and Var[Sn] and variance definition or formula. using the expectation b. Derive the generating function of this random walk Gs, (s). c. Use the generating function directly to calculate E[Sn] Var[Sn] and hence re-confirm the results in question la. , and d. Sketch the plot of the discrete pdf, and discuss its relation to the normal distribution for p = 0.5

Without using a moment generating function; Prove that the variance of a beta-distributed random variable with parameters α and β is σ2 = αβ/[(α + β)^2 (α + β + 1)]

NEED ANSWER FOR BOTH ASAP BOTH ARE FOR SAME QUESTION

Part II: Sections 2.1 - 2.7 8. Assume that X is a geometric random variable with p=0.32. (a) Compute P(X > 13[X > 3). (b) Compute E(X²).

Consider an AR(2) model: STEPS; -Write the equation of the AR(2) model- Determine the model based on the delay operator.- Find the expected value of the model- Find the variance of the model.- Find the covariances associated with 1,2, and s steps.- Find the associated correlation indices of 1,2, and s steps.- Assume that information is available up to time $h$ and the function that contains the accumulated information f(h, h-1,……), determine the forecasts and the error associated with 1,2, and s steps.- Assume that there is an initial value of the returns, denoted by $r_0$, and we have the AR(2) model at time rt, convert the AR(2) model into its equivalent MA(t)

Let X11 X12, Xini and X21, X22, X2n2 be two independent random samples of size n₁ and n₂ from two normal populations N(₁, 2) and N(2, 2) respectively. (a) Derive the maximum likelihood estimators (mle's) of all the parameters in the first population (X₁). Using analogy, state the mle's of the parameters of the second population. (b) Find the pooled estimator of the common variance when it is assumed that of = = ². Suggest an unbiased estimator of o². |