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- on any day, the number of accidents on the highway has a Poisson distribution with parameter A. The parameter A varies from day to day an is itself a random variable. Find the mean and variance of the number of accidents per day when A is uniformly distributed on (0,3)Consider a Poisson distribution with u = 4.6 Compute f(2) (ANSWER MUST BE 4 DIGITS)2.) Use the moment generating function you just proved to verify that for a Poisson distribution, E(X) = λ and Var(X) = λ.
- Let X1, X2 and X3 be independent and identically distributed random variables that follow a Poisson distribution with parameter k. Let Y = X1+X2+X3. Use the moment generating technique to determine the distribution of Y.Let X1, X2,..Xn be a random sample from a Poisson distribution with variance 0. Find the UMVUE for Pe(X=0). R(Ctrl) -Let X have a Poisson distribution. If , find the mode of the distribution. Suppose Y has a uniform pdf over the interval [3,5]. Find its (1) first quartile, its (2) median, and (3) the sum of its third quartile and its ninth decile.
- Find P(X=4) if X has a Poisson distribution such that 3P(X=1)=P(X=2)Let Y1, Y2, ... , Yn be a random sample of size n from a gamma distribution with parameters α = 1and β = 2. Derive the probability distribution of the sample mean Y̅ using moment-generatingfunctions.3 a. Let X have a Poisson distribution with a mean of 3. Define Y = (X-3) ^2 , specify pdf from Y.b. Let Y have a Poisson distribution with a mean of 5. Define Y=Z+X where Z and X are independent, where X is a random variable at point a, determine the distribution and pdf of Z.
- If the number X of particles emitted during a 1-h period from a radioactive source has a Poisson distribution with parameter 2 = 4 and that the probability that any emitted particle is recorded is p = 0.9, find the probability distribution of the number Y of the particles recorded in a 1-h period and hence the probability that no particle is recorded.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)=-1.53.Suppose that Xi,i =1,2,...,n have an Exponential distribution with parameter delta. a) Find maximum likelihood estimator for delta. b) is estimator is an unbiased estimator.