(9 points) In this question, we will find E(X), the nth moment of X,, where X,...
1. An application in probability (a) A function p(q) is a probability measure if p(x) > 0VT E R and (r) dx = 1. We first show that p(x):= vino exp(-) is a probability measure. (1) Compute dr. (ii) Show that were dr = 1. (ii) (1pt) Conclude that pr(I) is a probability measure. (b) A random variable x(): R + R is an integrable function that assigns a numerical value, X(I), to the outcome of an experiment, I, with...
Use integration to derive the moment-generating function MX (t) where fX (x) = (1/3) e^(−x/3) for x > 0. (Since we are maily interested in t near 0, assume that t < 1/3 .) Then use MX (t) to compute E(X), E(X^2), V (X), and E(X^3).
Exercise 5.22. Let X ~ Exp(A). Find the moment generating function of Y = ЗX — 2. Hint. Do not try to compute the probability density function of Y, instead use Mx (t)
12. let Mx(1) be the moment generating function of X. Show that (a) Mex+o(t) = eMx(at). (b) TX - Normal(), o?) and moment generating function of X is Mx (0) - to'p. Show that the random variable 2 - Normal(0,1) 13. IX. X X . are mutually independent normal random variables with means t o ... and variances o, o,...,0, then prove that X NOEL ?). 14. If Mx(1) be the moment generating function of X. Show that (a) log(Mx...
Given f(x) = ( c(x + 1) if 1 < x < 3 0 else as a probability function for a continuous random variable; find a. c. b. The moment generating function MX(t). c. Use MX(t) to find the variance and the standard deviation of X.
How to do (d) and (e)? Thanks. 11. Let X, X1, X2, ... be independent and identically distributed random variables taking values 0, 1, 2 with px(0) = 1, px(1) = 3 and px(2) = 1. Define Sn X1 Xn, n > 1. (a) Compute the probability generating function of X (b) Find the probability generating function of Sp. 2) from the probability generating function (c) Find P(Sn (d) Derive the moment generating function of S from its probability generating...
The moment generating function ф(t) of random variable X is defined for all values of t by et*p(x), if X is discrete e f (x)dx, if X is continus (a) Find the moment generating function of a Binomial random variable X with parameters n (the total number of trials) and p (the probability of success). (b) If X and Y are independent Binomial random variables with parameters (n1 p) and (n2, p), respectively, then what is the distribution of X...
9. Consider the Branching Process {Xn,n = 0,1,2,3,...} where Xn is the population size at the nth generation. Assume P(Xo = 1) = 1 and that the probability generating function of the offspring distribution is common A(z) (z3322z + 4) 10 (а) If gn 3 P(X, — 0) for n %3D 0, 1,..., write down an equation relating ^n+1 and qn. 0,1,2 Hence otherwise, evaluate qn for n= or (b) Find the extinction probability q = lim00 n 6 marks]...
2. Let Mx(t) = 1c' + 2t?c". Find the following: (b) Var(X). (c) If Y = X-2, show that the moment-generating function of Y is e-2tMx(t). (d) If W = 3X, show that the moment-generating function of W is MX(3). 7/3,5/9
(10 points) 4. The moment generating function of a random variable Y is , for t e R, where k is a constant. (a) Find the mean of Y. (b) Determine Pr(Y <1Y <2) (c) Find th e cumulative distribution function of Y, with domain R. (10 points) 4. The moment generating function of a random variable Y is , for t e R, where k is a constant. (a) Find the mean of Y. (b) Determine Pr(Y