![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](http://img.homeworklib.com/questions/473781e0-708c-11ea-8c6e-d737dcbc03b3.png?x-oss-process=image/resize,w_560)
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Problem 10: 10 points Assume that a random variable (L) follows the exponential distribution with intensity...
Problem 1: 10 points Assume that a random variable X follows the Poisson distribution with intensity-A,...
Problem 1: 10 points Assume that a random variable X follows the Poisson distribution with intensity-A, that is k! for k 0,1,2, . Using the identity (valid for all real t) exp(t) = Σ冠. k! k=0 derive the probability that X takes an even value, that is PIX is even
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
A discrete random variable A takes values {1, 2, 4} with probabilities specified as follows: P[A...
A discrete random variable A takes values {1, 2, 4} with probabilities specified as follows: P[A = 1] = 0.5, P[A = 2] = 0.3 and P [A = 4] = 0.2 Given A= ), a discrete random variable N is Poisson distributed with rate equal to 1, that is: 9 P[N = n|A = 1] = in n! el Hint If N is Poisson distributed with rate 1, its expectation and variance are as follows: E[N] = Var [N]...
[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...
[PLEASE USE HINT] Problem 4: 10 points Assume that a continuous random variable, Q, follows the...
[PLEASE USE HINT]
Problem 4: 10 points Assume that a continuous random variable, Q, follows the distribution, Beta [3,2], with the density function /9 (q) = 12q2 (1-1), Given Q = q, a random variable, X has the binomial distribution with n = 6, therefore for 0 < q < 1. 6! r! (6-2). g" (1-q)"-z for x 0, i, . . . , 6. 1. Derive the marginal expectation of X. 2. Derive the marginal variance of X Hint:...
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]
Problem The random variable X is exponential with parameter 1. Given the value r of X, the random variable Y is expo...
Problem The random variable X is exponential with parameter 1. Given the value r of X, the random variable Y is exponential with parameter equal to r (and mean 1/r) Note: Some useful integrals, for λ > 0: ar (a) Find the joint PDF of X and Y (b) Find the marginal PDF of Y (c) Find the conditional PDF of X, given that Y 2. (d) Find the conditional expectation of X, given that Y 2 (e) Find the...
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 ]
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...
X is a Poisson random variable of parameter 3 and Y an exponential random variable of...
X is a Poisson random variable of parameter 3 and Y an exponential random variable of parameter 3. Suppose X and Y are independent. Then A Var(2X + 9Y + 1) = 22 B Var(2X + 9Y + 1) = 7 CE[2X2 + 9Y2] = 19 D E[2X2 + 9Y2] = 26
Problem 1: 10 points Assume that a random variable X follows the Poisson distribution with intensity-A, that is k! for k 0,1,2, . Using the identity (valid for all real t) exp(t) = Σ冠. k! k=0 derive the probability that X takes an even value, that is PIX is even
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
A discrete random variable A takes values {1, 2, 4} with probabilities specified as follows: P[A = 1] = 0.5, P[A = 2] = 0.3 and P [A = 4] = 0.2 Given A= ), a discrete random variable N is Poisson distributed with rate equal to 1, that is: 9 P[N = n|A = 1] = in n! el Hint If N is Poisson distributed with rate 1, its expectation and variance are as follows: E[N] = Var [N]...
[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...
[PLEASE USE HINT]
Problem 4: 10 points Assume that a continuous random variable, Q, follows the distribution, Beta [3,2], with the density function /9 (q) = 12q2 (1-1), Given Q = q, a random variable, X has the binomial distribution with n = 6, therefore for 0 < q < 1. 6! r! (6-2). g" (1-q)"-z for x 0, i, . . . , 6. 1. Derive the marginal expectation of X. 2. Derive the marginal variance of X Hint:...
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]
Problem The random variable X is exponential with parameter 1. Given the value r of X, the random variable Y is exponential with parameter equal to r (and mean 1/r) Note: Some useful integrals, for λ > 0: ar (a) Find the joint PDF of X and Y (b) Find the marginal PDF of Y (c) Find the conditional PDF of X, given that Y 2. (d) Find the conditional expectation of X, given that Y 2 (e) Find 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 ]
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...
X is a Poisson random variable of parameter 3 and Y an exponential random variable of parameter 3. Suppose X and Y are independent. Then A Var(2X + 9Y + 1) = 22 B Var(2X + 9Y + 1) = 7 CE[2X2 + 9Y2] = 19 D E[2X2 + 9Y2] = 26
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