![Problem 7: 10 points Consider independent random variables, [Y: 1, 2, ..., ), having the same Gamma distribution, with the density, .узе-2y for y > 0 Suppose that a random variable, N, does not depend on all Y, and is geometrically distributed, so that PIN = n] = n, for n=1, 2, Consider a random sum, S Y 1. Determine the marginal expectation of S. 2. Determine the marginal variance of S.](http://img.homeworklib.com/questions/e28ec5d0-77bc-11ea-9fc7-6dfeff19f720.png?x-oss-process=image/resize,w_560)
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Problem 7: 10 points Consider independent random variables, [Y: 1, 2, ..., ), having the same...
You are given that a random variable, N, has the geometric distribution, 3-1 PIN = n]...
You are given that a random variable, N, has the geometric distribution, 3-1 PIN = n] = _ for n=1, 2, ,. Random variables, 偶; j=1, 2, . ) do not depend on N and are independent with the common exponential distribution, with the mean equal to θ 2, or equivalently (2)-, e-0.5x, for x 〉 0. Consider a random sum, 1. Derive the marginal expectation of S 2. Derive the marginal variance of S. 3. Find the marginal second...
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]
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 4 [10 points Assume that variables, (X1, X2, with the same Consider Y-Σ, xi. АЗ,...
Problem 4 [10 points Assume that variables, (X1, X2, with the same Consider Y-Σ, xi. АЗ, }, conditionally, given Q, are independent Bernoulli distributed parameter, Q. The marginal distribution of Q is uniform over the unit interval (o, Hint Use the identity (valid for integer a 20 and b 2 0): a! b! 1. Find marginal distribution of Y, for k 0,1,2,3. 2. Derive the conditional density for Q, given that Y -2 3. Derive conditional expectation and conditional variance...
Problem 5: 10 points Assume that a discrete random variable, N, is Poisson distributed with the...
Problem 5: 10 points Assume that a discrete random variable, N, is Poisson distributed with the rate, λ = 3. Given N = n, the random variable, X, conditionally has the binomial distribution, Bin [N +1, 0.4] 1. Evaluate the marginal expectation of X. 2. Evaluate the marginal variance of X
Suppose X, Y and Z are independent standard normal random variables. Then W = 2X +...
Suppose X, Y and Z are independent standard normal random variables. Then W = 2X + Y - Z is a random variable with mean 0 and variance 2, but not necessarily normal distributed. a normal random variable with mean 0 and variance 4. O a random variable with mean 0 and variance 4, but not necessarily normal distributed. a random variable with mean 0 and variance 6, but not necessarily normal distributed. a normal random variable with mean 0...
Suppose X, Y and Z are independent standard normal random variables. Then W = 2X +...
Suppose X, Y and Z are independent standard normal random variables. Then W = 2X + Y - Z is a random variable with mean 0 and variance 2, but not necessarily normal distributed. a normal random variable with mean 0 and variance 4. O a random variable with mean 0 and variance 4, but not necessarily normal distributed. a random variable with mean 0 and variance 6, but not necessarily normal distributed. a normal random variable with mean 0...
Problem 10: 10 points Assume that a random variable (L) follows the exponential distribution with intensity...
Problem 10: 10 points Assume that a random variable (L) follows the exponential distribution with intensity λ-1. Given L-u, a random variable Y has the Poisson distribution with parameter - u. 1. Derive the marginal distribution of Y and evaluate probabilities, PY=n] , for n = 0,1,2, 2. Find the expectation of Y, that is E Y 3. Find the variance of Y, that is Var Y
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 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....
You are given that a random variable, N, has the geometric distribution, 3-1 PIN = n] = _ for n=1, 2, ,. Random variables, 偶; j=1, 2, . ) do not depend on N and are independent with the common exponential distribution, with the mean equal to θ 2, or equivalently (2)-, e-0.5x, for x 〉 0. Consider a random sum, 1. Derive the marginal expectation of S 2. Derive the marginal variance of S. 3. Find the marginal second...
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]
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 4 [10 points Assume that variables, (X1, X2, with the same Consider Y-Σ, xi. АЗ, }, conditionally, given Q, are independent Bernoulli distributed parameter, Q. The marginal distribution of Q is uniform over the unit interval (o, Hint Use the identity (valid for integer a 20 and b 2 0): a! b! 1. Find marginal distribution of Y, for k 0,1,2,3. 2. Derive the conditional density for Q, given that Y -2 3. Derive conditional expectation and conditional variance...
Problem 5: 10 points Assume that a discrete random variable, N, is Poisson distributed with the rate, λ = 3. Given N = n, the random variable, X, conditionally has the binomial distribution, Bin [N +1, 0.4] 1. Evaluate the marginal expectation of X. 2. Evaluate the marginal variance of X
Suppose X, Y and Z are independent standard normal random variables. Then W = 2X + Y - Z is a random variable with mean 0 and variance 2, but not necessarily normal distributed. a normal random variable with mean 0 and variance 4. O a random variable with mean 0 and variance 4, but not necessarily normal distributed. a random variable with mean 0 and variance 6, but not necessarily normal distributed. a normal random variable with mean 0...
Suppose X, Y and Z are independent standard normal random variables. Then W = 2X + Y - Z is a random variable with mean 0 and variance 2, but not necessarily normal distributed. a normal random variable with mean 0 and variance 4. O a random variable with mean 0 and variance 4, but not necessarily normal distributed. a random variable with mean 0 and variance 6, but not necessarily normal distributed. a normal random variable with mean 0...
Problem 10: 10 points Assume that a random variable (L) follows the exponential distribution with intensity λ-1. Given L-u, a random variable Y has the Poisson distribution with parameter - u. 1. Derive the marginal distribution of Y and evaluate probabilities, PY=n] , for n = 0,1,2, 2. Find the expectation of Y, that is E Y 3. Find the variance of Y, that is Var Y
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 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....
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