Find a) the mean function for {x,}. b) the variance function for {x,}. c) the covariance for {x,}.
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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(2πω0hLet i_t denote the effective annual return achieved on an equity fund achieved between time (t -1) and time t. Annual log-returns on the fund, denoted by In(1 + i_t) , are assumed to form a series of independent and identically distributed Normal random variables with parameters u = 6% and o = 14%.An investor has a liability of £10,000 payable at time 15. Calculate the amount of money that should be invested now so that the probability that the investor will be unable to meet the liability as it falls due is only 5%. Using only formulas, no tablesSuppose that you have ten lightbulbs, that the lifetime ofeach is independent of all the other lifetimes, and that eachlifetime has an exponential distribution with parameter l.a. What is the probability that all ten bulbs fail beforetime t?b. What is the probability that exactly k of the ten bulbsfail before time t?c. Suppose that nine of the bulbs have lifetimes that areexponentially distributed with parameter l and thatthe remaining bulb has a lifetime that is exponentiallydistributed with parameter u (it is made byanother manufacturer). What is the probability thatexactly five of the ten bulbs fail before time t?
- Two resistors, with resistances R1 and R2, are connected in series. R1 is normally distributed with mean 100Ω and standard deviation 4Ω, and R2 is normally distributed with mean 120Ω and standard deviation 10Ω. What is the probability that R2 exceeds R1 by more than 30 Ω?Two resistors, with resistances R1 and R2, are connected in series. R1 is normally distributed with mean 100 Ω and standard deviation 5 Ω,and R2 is normally distributed with mean 120 Ω and standard deviation 10 Ω. a) What is the probability that R2 > R1? b) What is the probability that R2 exceeds R1 by more than 30 Ω.Consider 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 that X has a hypergeometric distribution with N = 500, n = 50, and K = 100. Find P(X = 10) by using the hypergeometric distribution and approximate binomial distribution. Is the binomial approximation reasonable?Find Taylor series at x = 0 for the functions cos (x5/3)Give power series f'(x).
- Suppose that x has a hypergeometric distribution with N = 300, n = 30, and K=100. 1. Find p(x=10) by using the hypergeometric distribution. 2. Find p(x=10) by using the approximate binomial distribution. 3. Is the binomial approximation reasonable?Let X1,X2,... be a sequence of identically distributed random variables with E|X1|<∞ and let Yn = n−1max1≤i≤n|Xi|. Show that limnE(Yn) = 0Define Power Series e^x?