![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: You will need to evaluate the first and second moments of Q](http://img.homeworklib.com/questions/7786a7b0-77f9-11ea-918a-bd87eb9c1627.png?x-oss-process=image/resize,w_560)
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Problem 4: 10 points Assume that a continuous random variable, Q, follows the...
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 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 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]
Assume that the density function for a continuous random variable, Y, is defined as fY(y) =...
Assume that the density function for a continuous random variable, Y, is defined as fY(y) = 9y. exp(-3y) for (y>0) and f'(y) = 0 elsewhere. Given Y = y, the conditional C.D.F. for X is FX\Y (x\Y) =P[X 5 X Y = y) = 1 – exp(-x •y) for (x > 0). Questions below are related to the marginal distribution for X. 1. Derive the density, f* (x). 2. Evaluate the expectation, E(X)
Q 2. The probability density function of the continuous random variable X is given by Shell,...
Q 2. The probability density function of the continuous random variable X is given by Shell, -<< 0. elsewhere. f(x) = {&e*, -40<3<20 (a) Derive the moment generating function of the continuous random variable X. (b) Use the moment generating function in (a) to find the mean and variance of X.
Consider a continuous random variable X with the following probability density function: Problem 2 (15 minutes)...
Consider a continuous random variable X with the following
probability density function:
Problem 2 (15 minutes) Consider a continuous random variable X with the following probability density function: f(x) = {& Otherwise ?' 10 otherwise? a. Is /(x) a well defined probability density function? b. What is the mathematical expectation of U (2) = x (the mean of X, )? c. What is the mathematical expectation of U(z) = (1 - 2 (the variance of X, oº)?
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 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...
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]...
Continuous random variable
(e) A continuous random variable X has the probability density function given by: f(x) = ( 2x/√ k for 0 ≤ x ≤ 2 0 otherwise. i. Show that the constant k equals 16. ii. Find the expected value of X. iii. Find the variance of X. iv. Derive the cumulative distribution function, F(x). v. Calculate P(X < 1 | X < 1.5)
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 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 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]
Assume that the density function for a continuous random variable, Y, is defined as fY(y) = 9y. exp(-3y) for (y>0) and f'(y) = 0 elsewhere. Given Y = y, the conditional C.D.F. for X is FX\Y (x\Y) =P[X 5 X Y = y) = 1 – exp(-x •y) for (x > 0). Questions below are related to the marginal distribution for X. 1. Derive the density, f* (x). 2. Evaluate the expectation, E(X)
Q 2. The probability density function of the continuous random variable X is given by Shell, -<< 0. elsewhere. f(x) = {&e*, -40<3<20 (a) Derive the moment generating function of the continuous random variable X. (b) Use the moment generating function in (a) to find the mean and variance of X.
Consider a continuous random variable X with the following
probability density function:
Problem 2 (15 minutes) Consider a continuous random variable X with the following probability density function: f(x) = {& Otherwise ?' 10 otherwise? a. Is /(x) a well defined probability density function? b. What is the mathematical expectation of U (2) = x (the mean of X, )? c. What is the mathematical expectation of U(z) = (1 - 2 (the variance of X, oº)?
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 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...
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]...
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