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1. (a) Let X ~ Poisson(1). Find its probability generating function (PGF) gx(s). Use the PGF...
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...
1. Let X have probability generating function Gx (s) and let un generating function U(s) of the sequence uo, u1, ... satisfies P(X > n). Show that the (1- s)U(s) = 1 - Gx(s), whenever the series defining these generating functions converge. 1. Let X have probability generating function Gx (s) and let un generating function U(s) of the sequence uo, u1, ... satisfies P(X > n). Show that the (1- s)U(s) = 1 - Gx(s), whenever the series defining...
Let Ņ, X1. X2, . . . random variables over a probability space It is assumed that N takes nonnegative inteqer values. Let Zmax [X1, -. .XN! and W-min\X1,... ,XN Find the distribution function of Z and W, if it suppose N, X1, X2, are independent random variables and X,, have the same distribution function, F, and a) N-1 is a geometric random variable with parameter p (P(N-k), (k 1,2,.)) b) V - 1 is a Poisson random variable with...
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,...
iid 14 marksAssume that e Denote T 4i Gamma(k, A) and X1,... , X,,e Poisson(0) (a) [4 marks Show that the posterior distribution of 0 is Gamma(nTk, n ). (b) [4 marks Find the probability function of the marginal distribution of Y = nX. (Note that the conditional distribution of on Y is not the same X1, ..., Xn.) as on iid 14 marksAssume that e Denote T 4i Gamma(k, A) and X1,... , X,,e Poisson(0) (a) [4 marks Show...
The moment generating function (MGF) for a random variable X is: Mx (t) = E[e'X]. Onc useful property of moment generating functions is that they make it relatively casy to compute weighted sums of independent random variables: Z=aX+BY M26) - Mx(at)My (Bt). (A) Derive the MGF for a Poisson random variable X with parameter 1. (B) Let X be a Poisson random variable with parameter 1, as above, and let y be a Poisson random variable with parameter y. X...
Solve # 8.16: Let S=X1 + X2 + ... XN be a random sum. Assume N is Poisson with parameter lambda and the Xk are i.i.d. exponential with parameter gamma. Furthermore, assume the Xk are independent of N. What are the mean and variance of S?
9 Let Xi, X2, ..., Xn be an independent trials process with normal density of mean 1 and variance 2. Find the moment generating function for (a) X (b) S2 =X1 + X2 . (c) Sn=X1+X2 + . . . + Xn. (d) An -Sn/n 9 Let Xi, X2, ..., Xn be an independent trials process with normal density of mean 1 and variance 2. Find the moment generating function for (a) X (b) S2 =X1 + X2 . (c)...
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...
Suppose X = Exp(1) and Y= -ln(x) (a)Find the cumulative distribution function of Y . (b) Find the probability density function of Y . (c) Let X1, X2, ... , Xk be i.i.d. Exp(1), and let Mk = max{X1,..... , Xk)(Maximum of X1, ..., Xk). Find the probability density function of Mk.(Hint: P(min(X1, X2, X3) > k) = P(X1 >= k, X2 >= k, X3 >= kq, how about max ?) (d) Show that as k → 00, the CDF...