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Problem 2: 10 points A random variable, Z, has the Gamma distribution with the density: and...
Assumptions from problem #2 Problem 3: 10 points Continue with the same assumptions as in Problem...
Assumptions from problem #2
Problem 3: 10 points Continue with the same assumptions as in Problem 2. Recall that a random variable, Z, has the Gamma distribution with the density: fz (z) = λ2 z exp[-λ z] for z > 0, and fz(z) = 0, elsewhere. Conditionally given Z = z, a random variable, U, is uniformly distributed over the interval, (0, z) 1. Find conditional expectation. EZIU = ul. 2. Find conditional variance, VARZİU-ul 3. Find conditional expectation, E...
Problem 6: 10 points Assume that X and Y are independent random variables uniformly distributed over...
Problem 6: 10 points Assume that X and Y are independent random variables uniformly distributed over the unit interval (0,1) 1. Define Z max (X. Y) as the larger of the two, Derive the C.DF. and density function for Z. 2. Define W min(X,Y) as the smaller of the two. Derive the C.D.F.and density function for W 3. Derive the joint density of the pair (W. Z). Specify where the density if positive and where it takes a zero value....
Problem 6: 10 points Assume that X and Y are independent random variables uniformly distributed over...
Problem 6: 10 points Assume that X and Y are independent random variables uniformly distributed over the unit interval (0,1) 1. Define Z-max (X, Y) as the larger of the two. Derive the C.D.F. and density function for Z. 2. Define Wmin (X, Y) as the smaller of the two. Derive the C.D.F. and density function for W 3. Derive the joint density of the pair (W, Z). Specify where the density if positive and where it takes a zero...
Problem 2: 20 points 10 5 + 5) A continuous random variable (Y) has a density,...
Problem 2: 20 points 10 5 + 5) A continuous random variable (Y) has a density, fY (3e-3V for y>0 and f () 0, elsewhere. Given Y y, a discrete random variable, N, is Poisson distributed with the rate equal to y TA 1. Derive the marginal distribution of N 2. Determine the marginal expectation of N, EIN 3. Determine the marginal variance of N, Var[N]
Need only parts 5 and 6 Problem 6: 10 points Assume that X and Y are...
Need only parts 5 and 6
Problem 6: 10 points Assume that X and Y are independent random variables uniformly distributed over the unit interval (0, 1) 1. Define Z = max (X, Y) as the larger of the two. Derive the CD. F. and density function for Z 2. Define W- min (X, Y) as the smaller of the two. Derive the C.D.F. and density function for W. 3. Derive the joint density of the pair (W, Z). Specify...
P7 continuous random variable X has the probability density function fx(x) = 2/9 if P.5 The...
P7
continuous random variable X has the probability density function fx(x) = 2/9 if P.5 The absolutely continuous random 0<r<3 and 0 elsewhere). Let (1 - if 0<x< 1, g(x) = (- 1)3 if 1<x<3, elsewhere. Calculate the pdf of Y = 9(X). P. 6 The absolutely continuous random variables X and Y have the joint probability density function fx.ya, y) = 1/(x?y?) if x > 1,y > 1 (and 0 elsewhere). Calculate the joint pdf of U = XY...
3. Let Xi,... , Xio be a random sample of size 10 from a gamma distribution...
3. Let Xi,... , Xio be a random sample of size 10 from a gamma distribution with α--3 and β 1/e. The prior distribution of θ is a gamma distribution with α-10 and B-2. Recall that the gamma density is given by elsewhere, (a) Find the posterior distribution of θ (b) If we observe 17, use the mean of the posterior distribution to give a point estimate of θ.
1. Let X1, X2, X3 be continuous random variables with joint probability density function 00 <...
1. Let X1, X2, X3 be continuous random variables with joint probability density function 00 < Xi < 00,i=1,2,3 Consider the transformation U-X1, V = X , W-XY + X + X (a) Find the joint pdf (probability density function) of U, V and W. (b) Find the marginal pdf of U, and hence find E(U) and Var(U) (c) Find the marginal pdf of W, and hence find E(W) and Var(W) (d) Find the conditional pdf of U given Ww,...
Let the conditional probability distribution of Y given π be elsewhere In this problem we will...
Let the conditional probability distribution of Y given π be elsewhere In this problem we will assume that π is a random variable and that the marginal distribution of π has a probability density function given by: f(n) = 0 elsewhere (a) Find the joint probability density function of Y and π, that is f(y, π). Please find the marginal probability distribution of Y, ). (c) Find the conditional distribution of f( y). (d) What is the mean and variance...
) Let X, Y be two random variables with the following properties. Y had density function...
) Let X, Y be two random variables with the following
properties. Y had
density function fY (y) = 3y
2
for 0 < y < 1 and zero elsewhere. For 0 < y < 1, given
Y = y, X
had conditional density function fX|Y (x | y) = 2x
y
2 for 0 < x < y and zero elsewhere.
(a) Find the joint density function fX,Y . Be precise about where
the values (x, y) are non-zero....
Assumptions from problem #2
Problem 3: 10 points Continue with the same assumptions as in Problem 2. Recall that a random variable, Z, has the Gamma distribution with the density: fz (z) = λ2 z exp[-λ z] for z > 0, and fz(z) = 0, elsewhere. Conditionally given Z = z, a random variable, U, is uniformly distributed over the interval, (0, z) 1. Find conditional expectation. EZIU = ul. 2. Find conditional variance, VARZİU-ul 3. Find conditional expectation, E...
Problem 6: 10 points Assume that X and Y are independent random variables uniformly distributed over the unit interval (0,1) 1. Define Z max (X. Y) as the larger of the two, Derive the C.DF. and density function for Z. 2. Define W min(X,Y) as the smaller of the two. Derive the C.D.F.and density function for W 3. Derive the joint density of the pair (W. Z). Specify where the density if positive and where it takes a zero value....
Problem 6: 10 points Assume that X and Y are independent random variables uniformly distributed over the unit interval (0,1) 1. Define Z-max (X, Y) as the larger of the two. Derive the C.D.F. and density function for Z. 2. Define Wmin (X, Y) as the smaller of the two. Derive the C.D.F. and density function for W 3. Derive the joint density of the pair (W, Z). Specify where the density if positive and where it takes a zero...
Problem 2: 20 points 10 5 + 5) A continuous random variable (Y) has a density, fY (3e-3V for y>0 and f () 0, elsewhere. Given Y y, a discrete random variable, N, is Poisson distributed with the rate equal to y TA 1. Derive the marginal distribution of N 2. Determine the marginal expectation of N, EIN 3. Determine the marginal variance of N, Var[N]
Need only parts 5 and 6
Problem 6: 10 points Assume that X and Y are independent random variables uniformly distributed over the unit interval (0, 1) 1. Define Z = max (X, Y) as the larger of the two. Derive the CD. F. and density function for Z 2. Define W- min (X, Y) as the smaller of the two. Derive the C.D.F. and density function for W. 3. Derive the joint density of the pair (W, Z). Specify...
P7
continuous random variable X has the probability density function fx(x) = 2/9 if P.5 The absolutely continuous random 0<r<3 and 0 elsewhere). Let (1 - if 0<x< 1, g(x) = (- 1)3 if 1<x<3, elsewhere. Calculate the pdf of Y = 9(X). P. 6 The absolutely continuous random variables X and Y have the joint probability density function fx.ya, y) = 1/(x?y?) if x > 1,y > 1 (and 0 elsewhere). Calculate the joint pdf of U = XY...
3. Let Xi,... , Xio be a random sample of size 10 from a gamma distribution with α--3 and β 1/e. The prior distribution of θ is a gamma distribution with α-10 and B-2. Recall that the gamma density is given by elsewhere, (a) Find the posterior distribution of θ (b) If we observe 17, use the mean of the posterior distribution to give a point estimate of θ.
1. Let X1, X2, X3 be continuous random variables with joint probability density function 00 < Xi < 00,i=1,2,3 Consider the transformation U-X1, V = X , W-XY + X + X (a) Find the joint pdf (probability density function) of U, V and W. (b) Find the marginal pdf of U, and hence find E(U) and Var(U) (c) Find the marginal pdf of W, and hence find E(W) and Var(W) (d) Find the conditional pdf of U given Ww,...
Let the conditional probability distribution of Y given π be elsewhere In this problem we will assume that π is a random variable and that the marginal distribution of π has a probability density function given by: f(n) = 0 elsewhere (a) Find the joint probability density function of Y and π, that is f(y, π). Please find the marginal probability distribution of Y, ). (c) Find the conditional distribution of f( y). (d) What is the mean and variance...
) Let X, Y be two random variables with the following
properties. Y had
density function fY (y) = 3y
2
for 0 < y < 1 and zero elsewhere. For 0 < y < 1, given
Y = y, X
had conditional density function fX|Y (x | y) = 2x
y
2 for 0 < x < y and zero elsewhere.
(a) Find the joint density function fX,Y . Be precise about where
the values (x, y) are non-zero....
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