Statement. For n events A, Az, ..., Ag, we have A₂, 'n' n (α) 0) P(24) 2 | > ΣΡ(4)-(n-1) i=1 η ομώνα της χ (1) - i=1 i=1
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- A12 and P(D) : 5 10 3 Find Events C and D are mutually exclusive such that, P(C) = (a) P(C') (b) P(CUD)Suppose that {u: t = ...,-2,-1,0,1,2,...} is an independent time series with mean zero, variance σ²= 9.0. Suppose that the time series {x+: t = ...,-2,-1,0,1,2,...} satisfies the equation: 2 X= 1.5 X-1 - .5 X-2 +3.0+u- U-1- 3 a) Identify the time series. b) c) d) i) Determine the autocovariance function and autocorrelation function of the time series X+ - Xt-1- Find the random shock form of the time series. Suppose that the first five observations of the time series are x₁ = 2.5, x2 = 4.0, X3 = 3.7, x4 =1.0 and x5 1.5. = Use these observations to compute prediction intervals for the next 5 observations. (Compute both 95% and 66.7% prediction limits) ii) If the sixth observation turns out to be x6 = 4.3 use this information to re-compute prediction intervals for the next 4 observations. (Compute both 95% and 66.7% prediction limits)
- (22) Let X be a random variable with p.d.f. 3k e 0(d) Experiment: measure the quantities a, b, c (in cm) of rainfall in a day at each of three weather stations A, B and C. Event: it is driest at station A. 2. Suppose we have 2=[0,9]={xR;0Consider the experiment consisting of an infinite sequence of coin flips. Then N = {(w1, w2, ..) : w;e {1,0} , i = 1, 2, ...} Xm Let denote the indicator function of the event that the Calculate nth flip is heads. lim inf E(X„) а) E(lim inf X„) b)Prove the statement2. The nonnegative integer-valued random variable X has generating function 1 gx (t) log - at k) for k 0,1,2,..., E(X), and where q is a positive constant. Determine P(X Var(X)Let X1,..., Xn be independent and identically distributed random variables with Var(X1) < ∞. Show that n 1 EjX;→p EX1. п(п + 1) j= Note. A simple way to solve a problem of showing Yn →p a is to establish limn EYn = a and limn Var(Yn) = 0.2. A random variable X has the discrete uniform distribution f(x; k) = {6. elsewhere. x = 1, 2, ..., k, 0, k Show that the moment-generating function of X is e'(1 – ekt) k(1 – et) Mx(t) -Let X be one observation from a population with pmf |x| f(x|0) = (9) ¹ (1 – 0)¹−|™|, x = −1, 0, 1; 0 ≤ 0 ≤ 1. 2 That is, X takes on values -1, 0, 1 with probabilities f(−1; 0) = ƒ(1; 0) = 0/2 and ƒ(0; 0) = (1 — 0). (a) Find the MLE of 0. =Prove the followingSEE MORE QUESTIONSRecommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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