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2. Let X1 and X2 be independent Poisson random variables with parameters λ1 and A2. Show...
3. (25 pts.) Let X1, X2, X3 be independent random variables such that Xi~ Poisson (A), i 1,2,3. Let N = X1 + X2+X3. (a)...
3. (25 pts.) Let X1, X2, X3 be independent random variables such that Xi~ Poisson (A), i 1,2,3. Let N = X1 + X2+X3. (a) What is the distribution of N? (b) Find the conditional distribution of (X1, X2, X3) | N. (c) Now let N, X1, X2, X3, be random variables such that N~ Poisson(A), (X1, X2, X3) | N Trinomial(N; pi,p2.ps) where pi+p2+p3 = 1. Find the unconditional distribution of (X1, X2, X3).
3. (25 pts.) Let X1,...
Let X1 and X2 be independent exponential random variables with parameters λ1 and λ2respectively. Find the...
Let X1 and X2 be independent exponential random variables with parameters λ1 and λ2respectively. Find the joint probability density function of X1 + X2 and X1 − X2.
Let Xi and X2 be independent Poisson random variables with means λ! and λ2. (a) Find...
Let Xi and X2 be independent Poisson random variables with means λ! and λ2. (a) Find the distribution of x, + x, (b) Compute the conditional distribution of Xi given that X + X.-n
Suppose X1 and X2 are iid Poisson(θ) random variables and let T = X1 + 2X2....
Suppose X1 and X2 are iid Poisson(θ) random variables and let T = X1 + 2X2. (a) Find the conditional distribution of (X1,X2) given T = 7. (b) For θ = 1 and θ = 2, respectively, calculate all probabilities in the above conditional distribution and present the two conditional distributions numerically.
Let X1 and X2 be independent gamma distribution random variables with gamma (a1,1) and gamma (a2,...
Let X1 and X2 be independent gamma distribution random variables with gamma (a1,1) and gamma (a2, 1). Find the marginal distributions of x1/(x1+x2) and x2/(x1+x2).
Question 1: Conditional of Poisson random variabless is Multinomial Lct X1,.... X% be independent random variables...
Question 1: Conditional of Poisson random variabless is Multinomial Lct X1,.... X% be independent random variables and suppose that X, ~ Poisson(Ii). What is the conditional distribution of (Xi, . . . , Xk) given that Σ_1 X,-n?
Let Xi Pn(2) and X2 Pn(5) be two independent random variables and it that y = Xi + X-Pn(7). is shown (a) Given Y-n, n 20, what are the possible values of X1? (b) Calculate the conditional distributio...
Let Xi Pn(2) and X2 Pn(5) be two independent random variables and it that y = Xi + X-Pn(7). is shown (a) Given Y-n, n 20, what are the possible values of X1? (b) Calculate the conditional distribution of Xi given Y-n for n 2 0.
Let Xi Pn(2) and X2 Pn(5) be two independent random variables and it that y = Xi + X-Pn(7). is shown (a) Given Y-n, n 20, what are the possible values of X1? (b)...
Problem 4 Let X and y be independent Poisson(A) and Poisson(A2) random variables, respectively. i. Write...
Problem 4 Let X and y be independent Poisson(A) and Poisson(A2) random variables, respectively. i. Write an expression for the PMF of Z -X + Y. i.e.. pz[n] for all possible n. ii. Write an expression for the conditional PMF of X given that Z-n, i.e.. pxjz[kn for all possible k. Which random variable has the same PMF, i.e., is this PMF that of a Bernoulli, binomial, Poisson, geometric, or uniform random variable (which assumes all possible values with equal...
6. Conditioning independent Poisson variables on their sum. Let N, be independent Poisson variables with parameters...
6. Conditioning independent Poisson variables on their sum. Let N, be independent Poisson variables with parameters di. Think of the N, as the number of points of a Poissson scatter in disjoint parts of the plane with areas li, where the mean intensity is one point per unit area. a) What is the conditional joint distribution of (N1, ... , Nm) given Ni+...+Nm = n? (Hint: See Exercise 5 for a special case.) b) Suppose now that N has Poisson(2)...
Let X1, X2, X3 be independent random variables with E(X1) = 1, E(X2) = 2 and...
Let X1, X2, X3 be independent random variables with E(X1) = 1, E(X2) = 2 and E(X3) = 3. Let Y = 3X1 − 2X2 + X3. Find E(Y ), Var(Y ) in the following examples. X1, X2, X3 are Poisson. [Recall that the variance of Poisson(λ) is λ.] X1, X2, X3 are normal, with respective variances σ12 = 1, σ2 = 3, σ32 = 5. Find P(0 ≤ Y ≤ 5). [Recall that any linear combination of independent normal...
3. (25 pts.) Let X1, X2, X3 be independent random variables such that Xi~ Poisson (A), i 1,2,3. Let N = X1 + X2+X3. (a) What is the distribution of N? (b) Find the conditional distribution of (X1, X2, X3) | N. (c) Now let N, X1, X2, X3, be random variables such that N~ Poisson(A), (X1, X2, X3) | N Trinomial(N; pi,p2.ps) where pi+p2+p3 = 1. Find the unconditional distribution of (X1, X2, X3).
3. (25 pts.) Let X1,...
Let Xi and X2 be independent Poisson random variables with means λ! and λ2. (a) Find the distribution of x, + x, (b) Compute the conditional distribution of Xi given that X + X.-n
Question 1: Conditional of Poisson random variabless is Multinomial Lct X1,.... X% be independent random variables and suppose that X, ~ Poisson(Ii). What is the conditional distribution of (Xi, . . . , Xk) given that Σ_1 X,-n?
Let Xi Pn(2) and X2 Pn(5) be two independent random variables and it that y = Xi + X-Pn(7). is shown (a) Given Y-n, n 20, what are the possible values of X1? (b) Calculate the conditional distribution of Xi given Y-n for n 2 0.
Let Xi Pn(2) and X2 Pn(5) be two independent random variables and it that y = Xi + X-Pn(7). is shown (a) Given Y-n, n 20, what are the possible values of X1? (b)...
Problem 4 Let X and y be independent Poisson(A) and Poisson(A2) random variables, respectively. i. Write an expression for the PMF of Z -X + Y. i.e.. pz[n] for all possible n. ii. Write an expression for the conditional PMF of X given that Z-n, i.e.. pxjz[kn for all possible k. Which random variable has the same PMF, i.e., is this PMF that of a Bernoulli, binomial, Poisson, geometric, or uniform random variable (which assumes all possible values with equal...
6. Conditioning independent Poisson variables on their sum. Let N, be independent Poisson variables with parameters di. Think of the N, as the number of points of a Poissson scatter in disjoint parts of the plane with areas li, where the mean intensity is one point per unit area. a) What is the conditional joint distribution of (N1, ... , Nm) given Ni+...+Nm = n? (Hint: See Exercise 5 for a special case.) b) Suppose now that N has Poisson(2)...
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