Let (N) (a) E(N3N4) (b) E(X3X4) (c) E(S3S4) 20 be a Poisson process with parameter A...
Let (N) (a) E(N3N4) (b) E(X3X4) (c) E(S3S4) 20 be a Poisson process with parameter A 2. Find the following:
Let {X(t); t >= 0) be a Poisson process having rate parameter lambda = 1. For the random variable, X(t), the number of events occurring in an interval of length t. Determine the following. (a) Pr(X(3.7) = 3|X(2.2) >= 2) (b) Pr(X(3.7) = 1|X(2.2) <2) (c) E(X(5)|X(10) = 7)
Let Y1,K,Y n denote a random sample from a Poisson distribution with parameter λ . a. Find a sufficient statistics for λ. b. Find the minimum variance unbiased estimator(MVUE) of λ2 .
> 0, that is 7. Let X has a Poisson distribution with parameter P(X = x) = e- Tendte 7. x = 0, 1, 2, .... Find the variance of X.
1. Since it is a stochastic Poisson process N (t) with parameter λ, what is the probability that N (t) is even? And odd?
9. Let X be a Poisson random variable with parameter k = 3. (a) P[X 25] (b) Find P[5 S X <10) (c) Find the variance ? 10. Use the related Table to find the following: (here Z represents the standard normal variable) (a) P[Z > 2.57] (b) The point z such that PL-2 SZ sz]=0.8
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Exercise 10.4. Let X be a Poisson random variable with parameter λ. That is, P(X = k) e-λλk/kl, k 0.1 Compute the characteristic function of (X-λ)/VA and find its limit as
Exercise 10.4. Let X be a Poisson random variable with parameter λ. That is, P(X = k) e-λλk/kl, k 0.1 Compute the characteristic function of (X-λ)/VA and find its limit as
Let X be a Poisson random variable with parameter λ = 6, and let Y = min(X, 12). (a) What is the p.m.f. of X? (b) What is the mean of X? (c) What is the variance of X? (d) What is the p.m.f. of Y? (e) Compute E[Y ].
Let N be a Poisson random variable with parameter λ. Suppose ξ1,ξ2,... is a sequence of i.i.d. random variables with mean µ and variance σ2, independent of N. Let SN = ξ1 + ...ξN. Determine the mean and variance of SN
1. Let (N(t))>o be a Poisson process with rate X, and let Y1,Y2, ... bei.i.d. random variables. Fur- ther suppose that (N(t))=>0 and (Y)>1 are independent. Define the compound Poisson process N(t) Y. X(t) = Recall that the moment generating function of a random variable X is defined by ºx(u) = E[c"X]. Suppose that oy, (u) < for all u CR (for simplicity). (a) Show that for all u ER, ºx() (u) = exp (Atløy, (u) - 1)). (b) Instead...