X1 and X2 are two independent random variables that have Poisson distributions with mean lambda1 and...

X1 and X2 are two independent random variables that have Poisson distributions with mean lambda1 and lambda2, respectively.

a) Use moment generating functions, derive and name the distribution of X = X1+X2

b) Derive and name the conditional distribution of X1 given that X = N where N is a fixed positive integer.

Please explain your answer in detail. Please don't copy other answers. Thank you; will thumb up!

Solutions

Expert Solution

orchestra answered 1 year ago

Next > < Previous

Related Solutions

Let X1, X2, . . . , Xn be iid Poisson random variables with unknown mean...

Let X1, X2, . . . , Xn be iid Poisson random variables with unknown mean µ 1. Find the maximum likelihood estimator of µ 2.Determine whether the maximum likelihood estimator is unbiased for µ

Let the independent random variables X1, X2, and X3 have binomial distributions with parameters n1=3, n2=5,...

Let the independent random variables X1, X2, and X3 have binomial distributions with parameters n1=3, n2=5, n3=2 and the same probabilitiy of success p = 2/5. Find P(X1=1-X3). Find P(X1=X3). Find P(X1+X2+X3>=1). Find the expected value and variance for X1+X2+X3.

Consider independent random variables X1, X2, and X3 such that X1 is a random variable having...

Consider independent random variables X1, X2, and X3 such that X1 is a random variable having mean 1 and variance 1, X2 is a random variable having mean 2 and variance 4, and X3 is a random variable having mean 3 and variance 9. (a) Give the value of the variance of X1 + (1/2)X2 + (1/3)X3 (b) Give the value of the correlation of Y = X1- X2 and Z = X2 + X3.

Suppose that X1 and X2 are two random variables. Suppose that X1 has mean 1 and...

Suppose that X1 and X2 are two random variables. Suppose that X1 has mean 1 and variance 4 while X2 has mean 3 and variance 9. Finally, suppose that the correlation between X1 and X2 is 3/8. Denote Y = 2X1 − X2. (67) The mean of Y is (a) 1 (b) 4 (c) -2 (d) -1 (68) The variance of Y is (a) 25 (b) 4 (c) 9 (d) 16 (69) The standard deviation of Y is (a) 5...

Suppose X1, X2, . . ., Xn are iid Poisson random variables with parameter λ. (a)...

Suppose X1, X2, . . ., Xn are iid Poisson random variables with parameter λ. (a) Find the MVUE for λ. (b) Find the MVUE for λ 2

Let X1, X2, . . . be a sequence of independent and identically distributed random variables...

Let X1, X2, . . . be a sequence of independent and identically distributed random variables where the distribution is given by the so-called zero-truncated Poisson distribution with probability mass function; P(X = x) = λx/ (x!(eλ − 1)), x = 1, 2, 3... Let N ∼ Binomial(n, 1−e^−λ ) be another random variable that is independent of the Xi ’s. 1) Show that Y = X1 +X2 + ... + XN has a Poisson distribution with mean nλ.

Let X1 and X2 be two independent random variables having a chi-squared distribution with degrees of...

Let X1 and X2 be two independent random variables having a chi-squared distribution with degrees of freedom n1 and n2, respectively. Let Y1 = (X1) / (X1 + X2) and Y2 = X1 + X2 (a) Find the joint p.d.f. of Y1 and Y2 (b) Find the marginal p.d.f. of each of Y1 and Y2 (c) Are Y1 and Y2 independent ? Justify your answer.

1. Consider the following weighted averages of independent random variables X1, X2, X3, all with mean...

1. Consider the following weighted averages of independent random variables X1, X2, X3, all with mean u and variance σ^2 θ1 = 1/3(X1) + 1/3(X2) + 1/3(X3) θ2 = 1/4(X1) + 2/4(X2) + 1/4(X3) θ3 = 2/5(X1) + 2/5(X2) + 2/5(X3) a) Find E[θ1], E[θ2], E[θ3] b) Are θ1, θ2 and θ3 unbiased for u? Explain c) Find the variance for θ1, θ2 and θ3 d) If you had to use one of the above estimators, which would you pick?...

(i) Find the marginal probability distributions for the random variables X1 and X2 with joint pdf...

(i) Find the marginal probability distributions for the random variables X1 and X2 with joint pdf f(x1, x2) = 12x1x2(1-x2) , 0 < x1 <1 0 < x2 < 1 , otherwise (ii) Calculate E(X1) and E(X2) (iii) Are the variables X1 and X2 stochastically independent? (iv) Given the variables in the question, find the conditional p.d.f. of X1 given 0<x2< ½ and the conditional expectation E[X1|0<x2< ½ ].

2. Let X1, X2, . . . , Xn be independent, uniformly distributed random variables on...

2. Let X1, X2, . . . , Xn be independent, uniformly distributed random variables on the interval [0, θ]. (a) Find the pdf of X(j) , the j th order statistic. (b) Use the result from (a) to find E(X(j)). (c) Use the result from (b) to find E(X(j)−X(j−1)), the mean difference between two successive order statistics. (d) Suppose that n = 10, and X1, . . . , X10 represents the waiting times that the n = 10...

Subjects