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It is known that for independent Poisson distributed random variables X1 Poisson(1) and X2
Poisson(2) it holds that X1 + X2 Poisson(1 + 2). Show that if Xi Poisson(i), i = 1, 2, . . . , k are
independent then the conditional distribution of X1 given X1 + X2 + . . .Xk is Binomial and
determine the parameters of this Binomial distribution. ii) Suppose that the X and Y are
components of continuous random vector with a density fX,Y (x, y) = cxy2, 0 < x < y, 0 < y < 2
(and zero else). Here c is a normalizing constant. a) Show that c = 5 16 . b) Find the marginal
density fX(x) and FX(x). c) Find the marginal density fY (y) and FY (y). d) Find the conditional
density fY |X(y|x). e) Find the conditional expected value a(x) = E(Y |X = x). Make sure that you
show your working and do not forget to always specify the support of the respective distribution

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It is known that for independent Poisson distributed random .pdf

  • 1. It is known that for independent Poisson distributed random variables X1 Poisson(1) and X2 Poisson(2) it holds that X1 + X2 Poisson(1 + 2). Show that if Xi Poisson(i), i = 1, 2, . . . , k are independent then the conditional distribution of X1 given X1 + X2 + . . .Xk is Binomial and determine the parameters of this Binomial distribution. ii) Suppose that the X and Y are components of continuous random vector with a density fX,Y (x, y) = cxy2, 0 < x < y, 0 < y < 2 (and zero else). Here c is a normalizing constant. a) Show that c = 5 16 . b) Find the marginal density fX(x) and FX(x). c) Find the marginal density fY (y) and FY (y). d) Find the conditional density fY |X(y|x). e) Find the conditional expected value a(x) = E(Y |X = x). Make sure that you show your working and do not forget to always specify the support of the respective distribution