1. Let X have probability generating function Gx (s) and let un generating function U(s) of...
1. (a) Let X ~ Poisson(1). Find its probability generating function (PGF) gx(s). Use the PGF to find EX (b) Let X1, ..., Xn be independent with marginal distribution Xk ~ Poisson(4x) for k = 1,..., n. Let S = X1 +...+ Xn denote the sum. Use PGFs to identify the distribution of
3. Use the probability generating function Px)(s) to find (a) E[X(10)] (b) VarX(10)] (c) P(X(5)-2) . ( 4.2 Probability Generating Functions The probability generating function (PGF) is a useful tool for dealing with discrete random variables taking values 0,1, 2, Its particular strength is that it gives us an easy way of characterizing the distribution of X +Y when X and Y are independent In general it is difficult to find the distribution of a sum using the traditional probability...
7. Let X a be random variable with probability density function given by -1 < x < 1 fx(x) otherwise (a) Find the mean u and variance o2 of X (b) Derive the moment generating function of X and state the values for which it is defined (c) For the value(s) at which the moment generating function found in part (b) is (are) not defined, what should the moment generating function be defined as? Justify your answer (d) Let X1,...
1 (10pts) Let U1, U2, ... ,Un be independent uniform random variables over [0, 0] with the probability density function (p.d.f). () = a 2 + [0, 03, 0 > 0. Let U(1), U(2), .-. ,U(n) be the order statistics. Also let X = U(1)/U(n) and Y = U(n)- (a) (5pts) Find the joint probability density function of (X, Y). (b) (5pts) From part (a), show that X and Y are independent variables.
Problem 7. Let U1,U2,... be independent random variables all uniformly distributed on the unit interval, and let N be the first integer n 2 2 such that Un > Un-1. Show that for each real number 0<u < 1 !-un . 1- e-". (a) P(Ui-u and N = n) = (b) PUI S u and N is even) Problem 7. Let U1,U2,... be independent random variables all uniformly distributed on the unit interval, and let N be the first integer...
Please answer d,e,f and g, thank you! roblem 1. Let (U common p.d.f. i 1 be a sequence of ii.d. discrete random variables with f(k) for k = 1, 2, 3 and for n 21 let Sn = Σ,u. (a) Find the probability that S2 is even. (b) Find the probability that Sn is even given that S,-1 is even. (e) Find the probability that Sn is even given that S-1 is odd. (d) Let pn P(Sn is even). Find...
Section 1.7: 4. Let f(x) be the exponential generating funcion of a sequence {%). Find the exponential generating functions for the follow- ing sequences in terms of f(x): (a) fan cl (b) foan (c (nani (e) 0, a,a, , (g) ao,0, a2,0, a,0,... (h) a, a2, a,... 8. (a) A sequence a satisfies the recurrence relation a3an+2, ao0 Find the exponential generating function ΣΧ0Lnz" Section 1.7: 4. Let f(x) be the exponential generating funcion of a sequence {%). Find the...
Assuming g is strictu increasing, prove un → u strongly in qe [1,p) given for ever (X, μ) be a measure space with finite rneasure. Let 1 < p < oo. let g : R → R be a continuous nondecreasing function such that for some constant C0 Define G(t) = Jog(s)ds. Let(%) be a sequence in LP(X) and let u E Lp(X) be such that and limsup G(un)du< where + = 1.
Verify the product rule for Formal Power Series. Very specifically: Let f(x) be the generating function for a sequence san) and g(x) be the generating function for a sequence sbn1. Using the definitions of multiplication and differentiation of FPS, write down a formula for the derivative of [f(x)g(x)]. Then write down a formula for f(x)g(x) + f(x)g'(x), where f(x) denotes the derivative of f(x). Then show that the two formulas describe the same formal power series. (i.e. both series have...
1. Consider a branchi ng process with offspring distribution (p) k2o- Let X be the number of individuals in the n-th generation. Let un- P(Xn - 0) be the probability that the population already dies out at generation n. Show that un-(-1) for all n2 2, where o is the probability generating function of the offspring distribution.