Can we find this without use of the moment
generating function?
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Can we find this without use of the moment
generating function?
3. (15 pts) Let Xị...
7. Let X1 and X2 be two iid exp(A) random variables. Set Yi Xi - X2...
7. Let X1 and X2 be two iid exp(A) random variables. Set Yi Xi - X2 and Y2 X + X2. Determine the joint pdf of Y and Y2, identify the marginal distributions of Yi and Y2, and decide whether or not Yi and Y2 are independent [10)
4. The moment generating function of the normal distribution with parameters μ and σ2 is (t)...
4. The moment generating function of the normal distribution with parameters μ and σ2 is (t) exp ( μ1+ σ2t2 ) for -oo < t oo. Show that E X)-ψ(0)-μ and Var(X)-ψ"(0)-[ty(0)12-σ2. 5. Suppose that X1, X2, and X3 are independent random variables such that E[X]0 and ElX 1 for i-12,3. Find the value of E[LX? (2X1 X3)2] 6. Suppose that X and Y are random variables such that Var(X)-Var(Y)-2 and Cov(X, Y)- 1. Find the value of Var(3X -...
7. Let X a be random variable with probability density function given by -1 < x < 1 fx(x) otherwise (a) Find the...
7. Let X a be random variable with probability density function given by -1 < x < 1 fx(x) otherwise (a) Find the mean u and variance o2 of X (b) Derive the moment generating function of X and state the values for which it is defined (c) For the value(s) at which the moment generating function found in part (b) is (are) not defined, what should the moment generating function be defined as? Justify your answer (d) Let X1,...
2. Let Z1 and Zo be independent standard normal random variables. Let! X= 221 +372 +12...
2. Let Z1 and Zo be independent standard normal random variables. Let! X= 221 +372 +12 X2 = 321 - 22 +11. (a) Find the joint density function of (X1, X2). (b) Find the covariance of X1 and X2. Now let Y1 = X1 + 4X2 +3 Y, = -2X2 +6X2 +5 (a) Find the joint density function of (Y1, Y). (b) Find the covariance of Yi and Y2.
12. (8 Pts.) Let Xi and X2 have the joint PDF Let Yi Xi/X2 and Y2...
12. (8 Pts.) Let Xi and X2 have the joint PDF Let Yi Xi/X2 and Y2 = Xy. Find the joint PDF of(H.)a). Are Y1 and Y2 independent?
2. The random variables X1, X2 and X3 are independent, with Xi N(0,1), X2 N(1,4) and...
2. The random variables X1, X2 and X3 are independent, with Xi N(0,1), X2 N(1,4) and X3 ~ N(-1.2). Consider the random column vector X-Xi, X2,X3]T. (a) Write X in the form where Z is a vector of iid standard normal random variables, μ is a 3x vector, and B is a 3 × 3 matrix. (b) What is the covariance matrix of X? (c) Determine the expectation of Yi = Xi + X3. (d) Determine the distribution of Y2...
Let X1 and X2 be two independent standard normal random variables. Define two new random variables...
Let X1 and X2 be two independent standard normal random variables. Define two new random variables as follows: Y-Xi X2 and Y2- XiBX2. You are not given the constant B but it is known that Cov(Yi, Y2)-0. Find (a) the density of Y (b) Cov(X2, Y2)
Problem 5 Let X and Y be random variables with joint PDF Px.y. Let ZX2Y2 and tan-1 (Y/X). Θ i. Find the joint PDF of Z and Θ in terms of the joint PDF of X and Y ii. Find the joint PDF of Z and Θ if...
Problem 5 Let X and Y be random variables with joint PDF Px.y. Let ZX2Y2 and tan-1 (Y/X). Θ i. Find the joint PDF of Z and Θ in terms of the joint PDF of X and Y ii. Find the joint PDF of Z and Θ if X and Y are independent standard normal random variables. What kind of random variables are Z and Θ? Are they independent?
Problem 5 Let X and Y be random variables with joint...
Let X1 and X2 be independent standard normal random vari ables, and let Y-AX-b, where Y-(y, Y)т, ...
Let X1 and X2 be independent standard normal random vari ables, and let Y-AX-b, where Y-(y, Y)т, X-(X1, X2)T, b (1, -2) and (a) Determine the joint pdf of ı and Y2 by using the formula given in class for the joint pdf of Y = g(X) when X and Y are random vectors of the same dimension, and q is invertible with both g and its inverse differentiable (b) Show that the joint pdf in (a) can be expressed...
Let X1, X2, ..., Xr be independent exponential random variables with parameter λ. a. Find the...
Let X1, X2, ..., Xr be independent exponential random variables with parameter λ. a. Find the moment-generating function of Y = X1 + X2 + ... + Xr. b. What is the distribution of the random variable Y?
7. Let X1 and X2 be two iid exp(A) random variables. Set Yi Xi - X2 and Y2 X + X2. Determine the joint pdf of Y and Y2, identify the marginal distributions of Yi and Y2, and decide whether or not Yi and Y2 are independent [10)
4. The moment generating function of the normal distribution with parameters μ and σ2 is (t) exp ( μ1+ σ2t2 ) for -oo < t oo. Show that E X)-ψ(0)-μ and Var(X)-ψ"(0)-[ty(0)12-σ2. 5. Suppose that X1, X2, and X3 are independent random variables such that E[X]0 and ElX 1 for i-12,3. Find the value of E[LX? (2X1 X3)2] 6. Suppose that X and Y are random variables such that Var(X)-Var(Y)-2 and Cov(X, Y)- 1. Find the value of Var(3X -...
7. Let X a be random variable with probability density function given by -1 < x < 1 fx(x) otherwise (a) Find the mean u and variance o2 of X (b) Derive the moment generating function of X and state the values for which it is defined (c) For the value(s) at which the moment generating function found in part (b) is (are) not defined, what should the moment generating function be defined as? Justify your answer (d) Let X1,...
2. Let Z1 and Zo be independent standard normal random variables. Let! X= 221 +372 +12 X2 = 321 - 22 +11. (a) Find the joint density function of (X1, X2). (b) Find the covariance of X1 and X2. Now let Y1 = X1 + 4X2 +3 Y, = -2X2 +6X2 +5 (a) Find the joint density function of (Y1, Y). (b) Find the covariance of Yi and Y2.
12. (8 Pts.) Let Xi and X2 have the joint PDF Let Yi Xi/X2 and Y2 = Xy. Find the joint PDF of(H.)a). Are Y1 and Y2 independent?
2. The random variables X1, X2 and X3 are independent, with Xi N(0,1), X2 N(1,4) and X3 ~ N(-1.2). Consider the random column vector X-Xi, X2,X3]T. (a) Write X in the form where Z is a vector of iid standard normal random variables, μ is a 3x vector, and B is a 3 × 3 matrix. (b) What is the covariance matrix of X? (c) Determine the expectation of Yi = Xi + X3. (d) Determine the distribution of Y2...
Let X1 and X2 be two independent standard normal random variables. Define two new random variables as follows: Y-Xi X2 and Y2- XiBX2. You are not given the constant B but it is known that Cov(Yi, Y2)-0. Find (a) the density of Y (b) Cov(X2, Y2)
Problem 5 Let X and Y be random variables with joint PDF Px.y. Let ZX2Y2 and tan-1 (Y/X). Θ i. Find the joint PDF of Z and Θ in terms of the joint PDF of X and Y ii. Find the joint PDF of Z and Θ if X and Y are independent standard normal random variables. What kind of random variables are Z and Θ? Are they independent?
Problem 5 Let X and Y be random variables with joint...
Let X1 and X2 be independent standard normal random vari ables, and let Y-AX-b, where Y-(y, Y)т, X-(X1, X2)T, b (1, -2) and (a) Determine the joint pdf of ı and Y2 by using the formula given in class for the joint pdf of Y = g(X) when X and Y are random vectors of the same dimension, and q is invertible with both g and its inverse differentiable (b) Show that the joint pdf in (a) can be expressed...
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