![Problem 1. 15 points] Let X be a uniform random variable in the interval [-1,2]. Let Y be an exponential random variable with mean 2. Assunne X and Y are independent. a) Find the joint sample space. b) Find the joint PDF for X and Y. c) Are X and Y uncorrelated? Justify your answer. d) Find the probability P1-1/4 < X < 1/2 1 Y < 21 e) Calculate E[X2Y2]](http://img.homeworklib.com/questions/09f38260-3b47-11eb-9a66-53a7fa5b5829.png?x-oss-process=image/resize,w_560)
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Problem 1. 15 points] Let X be a uniform random variable in the interval [-1,2]. Let...
Let X be a uniform(0, 1) random variable and let Y be uniform(1,2) with X and...
Let X be a uniform(0, 1) random variable and let Y be uniform(1,2) with X and Y being independent. Let U = X/Y and V = X. (a) Find the joint distribution of U and V . (b) Find the marginal distributions of U.
3. (25 points) Let be a uniform random variable on the interval (0,2m]. Define a cos...
3. (25 points) Let be a uniform random variable on the interval (0,2m]. Define a cos and y sin ?. Are x and y uncorrelated? Justify your answer
1. Let U be a random variable that is uniformly distributed on the interval (0,1) (a)...
1. Let U be a random variable that is uniformly distributed on the interval (0,1) (a) Show that V 1 - U is also a uniformly distributed random variable on the interval (0,1) (b) Show that X-In(U) is an exponential random variable and find its associated parameter (c) Let W be another random variable that is uformly distributed on (0,1). Assume that U and W are independent. Show that a probability density function of Y-U+W is y, if y E...
Q6 (4pt) Let X be a discrete uniform random variable over {1,2,...,6} and let Y be...
Q6 (4pt) Let X be a discrete uniform random variable over {1,2,...,6} and let Y be a Bernoulli random variable with parameter 1/2 such that X, Y are independent. (1) Find the PMF of the random variable Z, where Z XY. (2) Compute the third moment of Z, that is, E[z2
complete using R 3. Let X ~ Unif(1,2) be a uniform random variable on the interval...
complete using R
3. Let X ~ Unif(1,2) be a uniform random variable on the interval (1,2). (a) What is the exact value of the mean of X? (b) Compute or estimate the standard deviation of X. (c) Estimate the expected value E[1/X] accurately to two decimal places.
3. (Bpoints) Let X, Y and Z be independent uniform random variables on the interval (0, 2), Let W...
3. (Bpoints) Let X, Y and Z be independent uniform random variables on the interval (0, 2), Let W min(X, y.z a) Find pdf of W Find E(1-11 b)
3. (Bpoints) Let X, Y and Z be independent uniform random variables on the interval (0, 2), Let W min(X, y.z a) Find pdf of W Find E(1-11 b)
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...
Unif (0, 1) 5. Suppose U1 and U2 i= 1,2. Let X; = - log(1 -...
Unif (0, 1) 5. Suppose U1 and U2 i= 1,2. Let X; = - log(1 - U;), i = 1,2. [0, 1], U are independent uniform random variables on (a) Show that X1 and X2 are independent exponential random variables with mean 1, X; ~ Еxp(1), і — 1,2. (b) Find the joint density function of Y1 = X1 + X2 and Y2 = X1/X2 and show that Y1 and Y2 are independent.
Unif (0, 1) 5. Suppose U1 and...
2. Let X be a random variable that is uniform in (1,2) U (3,5). (a) Find...
2. Let X be a random variable that is uniform in (1,2) U (3,5). (a) Find the pdf and the cdf of X. (b) Compute the expectation of X. (c) Compute the variance of X. (d) Compute the skewness of X.
1. Let X be a random variable with variance ? > 0 and fx as a...
1. Let X be a random variable with variance ? > 0 and fx as a probability density function (pdf). The pdf is positive for all real numbers, that is fx(x) > 0. for all r ER Furthermore, the pdf fx is symmetric around zero, that is fx(x) = fx(-1), for all r ER Let y be the random variable given by Y = 4X2 +6X + with a,b,c E R. (i) For which values of a, b, and care...
3. (25 points) Let be a uniform random variable on the interval (0,2m]. Define a cos and y sin ?. Are x and y uncorrelated? Justify your answer
1. Let U be a random variable that is uniformly distributed on the interval (0,1) (a) Show that V 1 - U is also a uniformly distributed random variable on the interval (0,1) (b) Show that X-In(U) is an exponential random variable and find its associated parameter (c) Let W be another random variable that is uformly distributed on (0,1). Assume that U and W are independent. Show that a probability density function of Y-U+W is y, if y E...
Q6 (4pt) Let X be a discrete uniform random variable over {1,2,...,6} and let Y be a Bernoulli random variable with parameter 1/2 such that X, Y are independent. (1) Find the PMF of the random variable Z, where Z XY. (2) Compute the third moment of Z, that is, E[z2
complete using R
3. Let X ~ Unif(1,2) be a uniform random variable on the interval (1,2). (a) What is the exact value of the mean of X? (b) Compute or estimate the standard deviation of X. (c) Estimate the expected value E[1/X] accurately to two decimal places.
3. (Bpoints) Let X, Y and Z be independent uniform random variables on the interval (0, 2), Let W min(X, y.z a) Find pdf of W Find E(1-11 b)
3. (Bpoints) Let X, Y and Z be independent uniform random variables on the interval (0, 2), Let W min(X, y.z a) Find pdf of W Find E(1-11 b)
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...
Unif (0, 1) 5. Suppose U1 and U2 i= 1,2. Let X; = - log(1 - U;), i = 1,2. [0, 1], U are independent uniform random variables on (a) Show that X1 and X2 are independent exponential random variables with mean 1, X; ~ Еxp(1), і — 1,2. (b) Find the joint density function of Y1 = X1 + X2 and Y2 = X1/X2 and show that Y1 and Y2 are independent.
Unif (0, 1) 5. Suppose U1 and...
2. Let X be a random variable that is uniform in (1,2) U (3,5). (a) Find the pdf and the cdf of X. (b) Compute the expectation of X. (c) Compute the variance of X. (d) Compute the skewness of X.
1. Let X be a random variable with variance ? > 0 and fx as a probability density function (pdf). The pdf is positive for all real numbers, that is fx(x) > 0. for all r ER Furthermore, the pdf fx is symmetric around zero, that is fx(x) = fx(-1), for all r ER Let y be the random variable given by Y = 4X2 +6X + with a,b,c E R. (i) For which values of a, b, and care...
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