![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 U Z-z] 4. Find conditional variance. VARU -2]](http://img.homeworklib.com/questions/7312a490-6f61-11ea-ac28-2baf77871a07.png?x-oss-process=image/resize,w_560)
![Problem 2: 10 points A random variable, Z, has the Gamma distribution with the density: fz (z) λ2 z exp [-A z] for z > 0, and f ()0, elsewhere. According to the notation in Probability Theory, Z has the distribution Conditionally, given Z = z, a random variable, U, is uniformly distributed over the interval, (0,2). Gamma [2, λ-1]](http://img.homeworklib.com/questions/737e8f50-6f61-11ea-a197-6fd3194857c0.png?x-oss-process=image/resize,w_560)
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Assumptions from problem #2
Problem 3: 10 points Continue with the same assumptions as in Problem...
Problem 2: 10 points A random variable, Z, has the Gamma distribution with the density: and...
Problem 2: 10 points A random variable, Z, has the Gamma distribution with the density: and f ()0, elsewhere. According to the notation in Probability Theory, Z has the distribution Gamma [2. Conditionally, given Zz, a random variable, U, is uniformly distributed over the interval, (0,z) 1. Evaluate the joint density function of the pair, (Z, U). Indicate where this density is positive. 2. Derive the marginal density, fU (u) 3. Find the conditional density of Z, given Uu. Indicate...
Problem 7: 10 points Assume that a lifetime random variable (T) is exponentially distributed with the...
Problem 7: 10 points Assume that a lifetime random variable (T) is exponentially distributed with the intensity λ > 0. I. Determine conditional density of the residual lifetime, T-u, given that T 〉 u. 2. Find conditional expectation, E TT>u
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 7: 10 points Consider independent random variables, [Y: 1, 2, ..., ), having the same...
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.
Problem 7: 10 points Assume that a lifetime random variable (T) is exponentially distributed with the...
Problem 7: 10 points Assume that a lifetime random variable (T) is exponentially distributed with the intensity λ > 0. 1. Determine conditional density of the residual lifetime, T-u, given that T>u 2. Find conditional expectation, EITIT>] Solution
[Problem 1 Information] Problem 2: 10 points Continue with the Poisson distribution for X from...
[Problem 1 Information]
Problem 2: 10 points Continue with the Poisson distribution for X from Problem 1. Find the conditional expectation of X given that X takes an even value. oution for X from Problem 1. Find Assume that a random variable X follows the Poisson distribution with intensity λ, that is for k 0,1,2, . Using the identity (valid for all real t) k! k=0 derive the probability that X takes an even value, that is PX is...
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
Problem 8: 10 points Assume that a lifetime random variable (T) is exponentially distributed with the...
Problem 8: 10 points Assume that a lifetime random variable (T) is exponentially distributed with the intensity λ 〉 0. 1. Find conditional variance, Var TIT〉 u] . 2. Find conditional second moment, E T IT ]
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
In the above problem, what are the natural places where you would put the hospital? Why? Now, str...
In the above problem, what are the natural places where you
would put the hospital? Why? Now, strengthen your intuition by
determining the expected distance between the random accident point
and the fixed point a where the hospital is located. Find the value
of for which this expectation is minimized. Then look at your
first, intuitive answer. What have you learned?
PROBLEM 2 (13 points) An ambulance travels back and forth, at a constant speed, along a road of length...
Problem 2: 10 points A random variable, Z, has the Gamma distribution with the density: and f ()0, elsewhere. According to the notation in Probability Theory, Z has the distribution Gamma [2. Conditionally, given Zz, a random variable, U, is uniformly distributed over the interval, (0,z) 1. Evaluate the joint density function of the pair, (Z, U). Indicate where this density is positive. 2. Derive the marginal density, fU (u) 3. Find the conditional density of Z, given Uu. Indicate...
Problem 7: 10 points Assume that a lifetime random variable (T) is exponentially distributed with the intensity λ > 0. I. Determine conditional density of the residual lifetime, T-u, given that T 〉 u. 2. Find conditional expectation, E TT>u
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 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.
Problem 7: 10 points Assume that a lifetime random variable (T) is exponentially distributed with the intensity λ > 0. 1. Determine conditional density of the residual lifetime, T-u, given that T>u 2. Find conditional expectation, EITIT>] Solution
[Problem 1 Information]
Problem 2: 10 points Continue with the Poisson distribution for X from Problem 1. Find the conditional expectation of X given that X takes an even value. oution for X from Problem 1. Find Assume that a random variable X follows the Poisson distribution with intensity λ, that is for k 0,1,2, . Using the identity (valid for all real t) k! k=0 derive the probability that X takes an even value, that is PX is...
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
Problem 8: 10 points Assume that a lifetime random variable (T) is exponentially distributed with the intensity λ 〉 0. 1. Find conditional variance, Var TIT〉 u] . 2. Find conditional second moment, E T IT ]
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
In the above problem, what are the natural places where you
would put the hospital? Why? Now, strengthen your intuition by
determining the expected distance between the random accident point
and the fixed point a where the hospital is located. Find the value
of for which this expectation is minimized. Then look at your
first, intuitive answer. What have you learned?
PROBLEM 2 (13 points) An ambulance travels back and forth, at a constant speed, along a road of length...
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