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Question 3: Suppose that X and Y are i.i.d. N(0,1) r.v.'s (a)...
Question 2: Let X and Y be i.i.d. r.v.'s with common pdf (de-*, f(t) -At t...
Question 2: Let X and Y be i.i.d. r.v.'s with common pdf (de-*, f(t) -At t 2 0, - otherwise (a) (6 marks). Find the joint pdf for U X/(X + Ү) аnd V — X + Y. (b) (2 marks). Find fu(u) and fv(v) (c) (2 marks). Are U and V independent? Why?
Problem 5.1 (Relation between Gaussian and exponential) Suppose that Xi and X, are i.i.d. N(0,1) (a)...
Problem 5.1 (Relation between Gaussian and exponential) Suppose that Xi and X, are i.i.d. N(0,1) (a) Show that Z-X1 + X is exponential with mean 2. b) True or False: Z is independent of Θ-tan ( -i Hint: Use the results from Example 5.4.3, which tells us the joint distribution of V and Θ.
Suppose X, Y are independent with X ∼ N (0, 1) and Y ∼ N (0,...
Suppose X, Y are independent with X ∼ N (0, 1) and Y ∼ N (0, 1). Show that the distribution of Q = X/Y follows the Cauchy distribution, i.e., f(q) = 1/π(1+q2) . Hint: Let Q = X/Y and V=Y. Find the joint pdf of Q and V and finally find the marginal pdf of Q by integrating the joint pdf of Q and V w.r.t. V: Y π(1+q2) Y V = Y . Find the joint pdf of...
Let X have the pdf defined for 0<x<2. Let Y~Unif(0,1). Suppose X and Y are independent....
Let X have the pdf defined for 0<x<2. Let Y~Unif(0,1). Suppose X and Y are independent. Find the distribution of X-Y. fx() =
Suppose X and Y are iid Uniform[0,1] random variables. Please explain in detail how you get...
Suppose X and Y are iid Uniform[0,1] random variables.
Please explain in detail how you get the answer for each
question. Thanks.
(7) Suppose X and Y are iid Uniform[0,1random variables. Let U = X and(X the correct answer in each of parts (a), (b), (d), (e) and show your' work in part (c) Circle (а) Р(V - U < 1/2) %3 Jacobjan factor 1/2. 1/8 0. (b) The domain D where the joint density f(U,v(u, v) is defined is...
Suppose (X,Y ) is chosen according to the continuous uniform distribution on the triangle with vertices (0,0), (0,1) and (2,0), that is, the joint pdf of (X,Y ) is fX,Y (x,y) =c, for 0 ≤ x ≤ 2,0 ≤ y ≤...
Suppose (X,Y ) is chosen according to the continuous uniform distribution on the triangle with vertices (0,0), (0,1) and (2,0), that is, the joint pdf of (X,Y ) is fX,Y (x,y) =c, for 0 ≤ x ≤ 2,0 ≤ y ≤ 1, 1/ 2x + y ≤ 1, 0 , else. (a) Find the value of c. (b) Calculate the pdf, the mean and variance of X. (c) Calculate the pdf and the mean of Y . (d) Calculate the...
Show working out for all parts please. 15. Suppose the random variables X and Y have...
Show working out for all parts please.
15. Suppose the random variables X and Y have the following joint probability density function: 1/4 0< < 2y <4, fx.x(x,y) = { 0 otherwise. (Remember to justify your answers: see instructions at the top of the previous page.) (a) Set up the calculation required to find E[XY). If your expression contains an integral or a sum, do not evaluate it (b) What is fyx-2(y)? (e) Find P{X > a} for some real...
5. Let X have a uniform distribution on the interval (0,1). Given X = x, let...
5. Let X have a uniform distribution on the interval (0,1). Given X = x, let Y have a uniform distribution on (0, 2). (a) The conditional pdf of Y, given that X = x, is fyıx(ylx) = 1 for 0 < y < x, since Y|X ~U(0, X). Show that the mean of this (conditional) distribution is E(Y|X) = , and hence, show that Ex{E(Y|X)} = i. (Hint: what is the mean of ?) (b) Noting that fr\x(y|x) =...
3. Suppose X and Y have joint density f(x,y)- "cy. 0 < x < y <...
3. Suppose X and Y have joint density f(x,y)- "cy. 0 < x < y < oo, and equal to 0 for all other (r, y). (a) Calculate the joint density of U = Y-X,V-X. (b) Are U and V independent?
2. Suppose that (X,Y) has the following joint probability density function: f(x,y) = C if -1...
2. Suppose that (X,Y) has the following joint probability density function: f(x,y) = C if -1 <r< 1 and -1 <y<1, and 0 otherwise. Here is a constant. (a) Determine the value of C. (b) Are X and Y independent? (Explain why or why not.) (c) Calculate the probability that 2X - Y > 0 (d) Calculate the probability that |X+Y| < 2 3. Suppose that X1 and X2 are independent and each is standard uniform on (0,1]. Let Y...
Question 2: Let X and Y be i.i.d. r.v.'s with common pdf (de-*, f(t) -At t 2 0, - otherwise (a) (6 marks). Find the joint pdf for U X/(X + Ү) аnd V — X + Y. (b) (2 marks). Find fu(u) and fv(v) (c) (2 marks). Are U and V independent? Why?
Problem 5.1 (Relation between Gaussian and exponential) Suppose that Xi and X, are i.i.d. N(0,1) (a) Show that Z-X1 + X is exponential with mean 2. b) True or False: Z is independent of Θ-tan ( -i Hint: Use the results from Example 5.4.3, which tells us the joint distribution of V and Θ.
Suppose X and Y are iid Uniform[0,1] random variables.
Please explain in detail how you get the answer for each
question. Thanks.
(7) Suppose X and Y are iid Uniform[0,1random variables. Let U = X and(X the correct answer in each of parts (a), (b), (d), (e) and show your' work in part (c) Circle (а) Р(V - U < 1/2) %3 Jacobjan factor 1/2. 1/8 0. (b) The domain D where the joint density f(U,v(u, v) is defined is...
Show working out for all parts please.
15. Suppose the random variables X and Y have the following joint probability density function: 1/4 0< < 2y <4, fx.x(x,y) = { 0 otherwise. (Remember to justify your answers: see instructions at the top of the previous page.) (a) Set up the calculation required to find E[XY). If your expression contains an integral or a sum, do not evaluate it (b) What is fyx-2(y)? (e) Find P{X > a} for some real...
5. Let X have a uniform distribution on the interval (0,1). Given X = x, let Y have a uniform distribution on (0, 2). (a) The conditional pdf of Y, given that X = x, is fyıx(ylx) = 1 for 0 < y < x, since Y|X ~U(0, X). Show that the mean of this (conditional) distribution is E(Y|X) = , and hence, show that Ex{E(Y|X)} = i. (Hint: what is the mean of ?) (b) Noting that fr\x(y|x) =...
3. Suppose X and Y have joint density f(x,y)- "cy. 0 < x < y < oo, and equal to 0 for all other (r, y). (a) Calculate the joint density of U = Y-X,V-X. (b) Are U and V independent?
2. Suppose that (X,Y) has the following joint probability density function: f(x,y) = C if -1 <r< 1 and -1 <y<1, and 0 otherwise. Here is a constant. (a) Determine the value of C. (b) Are X and Y independent? (Explain why or why not.) (c) Calculate the probability that 2X - Y > 0 (d) Calculate the probability that |X+Y| < 2 3. Suppose that X1 and X2 are independent and each is standard uniform on (0,1]. Let Y...
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