2. A random variable X has the discrete uniform distribution i x=1,2,..., k, 10, elsewhere. f(x; k) = Show that the moment-generating function of X is e*(1 – e*t) k(1 – et) Мx() — |
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- 1 E (t) Suppose that X and Y are independent, X-B(10, 0.5), Y follows Poisson distribution with parameter 4. Which of the following statement is not true? A. E(X+Y)=9 B. E(X-Y)=1 C. D(X+Y)=6.5 D. D(X-Y)=-1.5A random walk (RW) {Sn}n>o is a sum of id r.v.s X1, X2, •• , Xn,withP(X1 = a) = p, P(X1 = b) = 1 -p= q.Find the expectation E(Snl and variance var(Sn) of Sn for any n.Let W₁ < W₂ < ... < Wn be the order statistics of n independent observations from a U(0, 1) distribution. (a) Find the pdf of W₁ and that of Wn. (b) Use the results of (a) to verify that E (W₁) = 1/(n+1) and E(Wn) = n/(n+1).
- (24) Let X be a random variable with p.d.f. Ze-2 0Let X1,..., Xn be a random sample from a uniform distribution on the interval [20, 0], where 0 0. Let X(1) < X(2) <...< X(n) be the order statistics of X1, ..., Xn.der a random sample X1, X2, ..., X, having the pdf, f(r; 0) 1Let X₁, X2, X3, be a random sample from a discrete distribution with probability function p(x) = for x = 0, for x = 1, otherwise. Determine the moment generating function M(t) of Y = X₁X₂X3- A. exp(t) B. (exp(t)+7)/8 C. (exp(1/2)+1)/3 D. (exp(t)+63)/64 E. (exp(t)+1)/4Let Y1, Y2,., Ya be a collection of independent random variables with distribution function y 8 Show that Y converges in probability to a constant, and provide that constant. 1Suppose Y is a continuous random variable drawn from the uniform distributionon the interval [3, 4], that is, Y ∼ Uniform([3, 4]). Conditioned on Y = y, a second randomvariable X is drawn from the uniform distribution on the interval [0, y]. What is fX(x), thepdf of X?Recommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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