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Answer true or false to the following statements. Justify your answer. [5pts per question (a) If...
Let X and Y be independent normal random variables with parameters E[X] =ux, E[Y] = uy...
Let X and Y be independent normal random variables with parameters E[X] =ux, E[Y] = uy and Var(X) = x, Var(Y) = Oy. Indicate whether each of the following statements is true or false. Notation: fx,y (x, y), fx(x), fy (v) denote the joint and marginal PDFs of X and Y , respectively; $(x) is the CDF of a standard normal random variable with zero mean and unit variance. E[XY]=0
(6 points) Let X and Y be independent random variables with p.d.f.s fx(x) -{ { 1-22...
(6 points) Let X and Y be independent random variables with p.d.f.s fx(x) -{ { 1-22 0, for |2|<1, otherwise. fy(y) = for y>0, otherwise. 0, Let W = XY (a) (2 points) Find the p.d.f. of W, fw(w). (b) (2 points) Find the moment generating function of W2, Mw?(t) = E (e«w?). (c) (2 points) Find the conditional expectation of W given Y = y, E(W|Y = y).
1. The joint probability density function (pdf) of X and Y is given by fxy(x, y)...
1. The joint probability density function (pdf) of X and Y is given by fxy(x, y) = A (1 – xey, 0<x<1,0 < y < 0 (a) Find the constant A. (b) Find the marginal pdfs of X and Y. (c) Find E(X) and E(Y). (d) Find E(XY). 2. Let X denote the number of times (1, 2, or 3 times) a certain machine malfunctions on any given day. Let Y denote the number of times (1, 2, or 3...
Write clearly and neatly your final answers in the table below: # Points Question Answer Plot...
Write clearly and neatly your final answers in the table below: # Points Question Answer Plot Vx, x and x2 Show your plot here 1- 5 Bonus 2- 10 Determinec Plot fxy(x,y) Show your plot here 3- 5 Bonus 4- 10P 6- 10 7- 10 8-15 5 0<x<0.25, 0 <Y < 0.5) P(Y<X fx(x) EX fry) Are X and Y independent? E(Y) E(XY) VOX) V(Y) cov(X,Y) 10 10 Show your detailed solutions below Given the function fxy(x, y) = cxy...
1. The Weibull distribution has many applications in reliability engineering, survival analysis, and general insurance. function...
1. The Weibull distribution has many applications in reliability engineering, survival analysis, and general insurance. function Let β > 0, δ > O. Consider the probability density fx(x) = βδΧ6-1 e-Px®, x>0, zero otherwise. Find the probability distribution of W-X" b) Determine the probability distribution of W by finding the p.d.f. of W, fw(w), using the change-of-variable technique D) Find the p.d.f. of W, w)-x(w ii What is the name of the probability distribution of W? What are its parameters?...
Problem 8. True or False (You do not need to explain your answer but must indicate...
Problem 8. True or False (You do not need to explain your answer but must indicate clearly “T” or “F.” You will lose points in your T or F is not clearly indicated.) (a)_ Given z=f(x,y), fxy(x, y) is always equal to fxx (x, y). (b)_ Tangent plane to z= f(x, y) at (x,,y.) always approximates the value of z near (xo, y.). (c) __ If fx and f, exists, then z= f(x,y) is differentiable. (d) __If f{(1,2)=0, then fx,...
Show all work! Thank you! Sk(x+y) 0<x<1, 0<y</ 14. Determine k, so that fx.y(x, y)= otherwise...
Show all work! Thank you!
Sk(x+y) 0<x<1, 0<y</ 14. Determine k, so that fx.y(x, y)= otherwise is a joint pdf. 10 15. Determine k, so that fxy(x,y)= kry 0<x<1, 0<y<1. 6 otherwise is a joint pdf. k(xy?) 0<x<1, 0<y<1. is a joint pdf. Determine k, so that fx.x(x,y)= 1 otherwise 17. Determine k, so that fx.y(x,y)= kr 0<x<y<1 O otherwise is a joint pdf. k(x + y) 0<x< y<1 18. Determine k, so that fx. (x,y)= 1 0 otherwise is...
6. Justify briefly (max two lines) if the following statements are true or false. Answers with...
6. Justify briefly (max two lines) if the following statements are true or false. Answers with just Tor F won't receive any points. (a) (1 point) The curve with vector (Cartesian) equation r(t) = (t cost, tsint, t) lies on the surface with cylindrical equation 2 = 72 (b) (1 point) If ñ x p=0 then ñ = 0 or p=0. (c) (1 point) 24 – 4y2 lim (x,y)+(0,0) 22 + 2y2 = 0 (d) (1 point) If f(x, y)...
3. (a) The bus 500 arrives at Liverpool Airport at a rate of A buses per...
3. (a) The bus 500 arrives at Liverpool Airport at a rate of A buses per hour. Assume that the arrivals form a Poisson process. Let X (t) be the number of buses that arrive in t hours. X(t) is distributed as Px(o(u)=e-Ar (Xt)" u! when u is a positive integer and 0 otherwise. Let Y be the amount of time that you must wait for the 3rd bus to arrive. The event X (t) < 3 (fewer than three...
Mark each statement as True or False and justify your answer. a) The columns of a...
Mark each statement as True or False and justify your answer. a) The columns of a matrix A are linearly independent, if the equation Ax = 0 has the trivial solution. b) If vi, i = 1, ...,5, are in RS and V3 = 0, then {V1, V2, V3, V4, Vs} is linearly dependent. c) If vi, i = 1, 2, 3, are in R3, and if v3 is not a linear combination of vi and v2, then {V1, V2,...
Let X and Y be independent normal random variables with parameters E[X] =ux, E[Y] = uy and Var(X) = x, Var(Y) = Oy. Indicate whether each of the following statements is true or false. Notation: fx,y (x, y), fx(x), fy (v) denote the joint and marginal PDFs of X and Y , respectively; $(x) is the CDF of a standard normal random variable with zero mean and unit variance. E[XY]=0
(6 points) Let X and Y be independent random variables with p.d.f.s fx(x) -{ { 1-22 0, for |2|<1, otherwise. fy(y) = for y>0, otherwise. 0, Let W = XY (a) (2 points) Find the p.d.f. of W, fw(w). (b) (2 points) Find the moment generating function of W2, Mw?(t) = E (e«w?). (c) (2 points) Find the conditional expectation of W given Y = y, E(W|Y = y).
1. The joint probability density function (pdf) of X and Y is given by fxy(x, y) = A (1 – xey, 0<x<1,0 < y < 0 (a) Find the constant A. (b) Find the marginal pdfs of X and Y. (c) Find E(X) and E(Y). (d) Find E(XY). 2. Let X denote the number of times (1, 2, or 3 times) a certain machine malfunctions on any given day. Let Y denote the number of times (1, 2, or 3...
Write clearly and neatly your final answers in the table below: # Points Question Answer Plot Vx, x and x2 Show your plot here 1- 5 Bonus 2- 10 Determinec Plot fxy(x,y) Show your plot here 3- 5 Bonus 4- 10P 6- 10 7- 10 8-15 5 0<x<0.25, 0 <Y < 0.5) P(Y<X fx(x) EX fry) Are X and Y independent? E(Y) E(XY) VOX) V(Y) cov(X,Y) 10 10 Show your detailed solutions below Given the function fxy(x, y) = cxy...
1. The Weibull distribution has many applications in reliability engineering, survival analysis, and general insurance. function Let β > 0, δ > O. Consider the probability density fx(x) = βδΧ6-1 e-Px®, x>0, zero otherwise. Find the probability distribution of W-X" b) Determine the probability distribution of W by finding the p.d.f. of W, fw(w), using the change-of-variable technique D) Find the p.d.f. of W, w)-x(w ii What is the name of the probability distribution of W? What are its parameters?...
Problem 8. True or False (You do not need to explain your answer but must indicate clearly “T” or “F.” You will lose points in your T or F is not clearly indicated.) (a)_ Given z=f(x,y), fxy(x, y) is always equal to fxx (x, y). (b)_ Tangent plane to z= f(x, y) at (x,,y.) always approximates the value of z near (xo, y.). (c) __ If fx and f, exists, then z= f(x,y) is differentiable. (d) __If f{(1,2)=0, then fx,...
Show all work! Thank you!
Sk(x+y) 0<x<1, 0<y</ 14. Determine k, so that fx.y(x, y)= otherwise is a joint pdf. 10 15. Determine k, so that fxy(x,y)= kry 0<x<1, 0<y<1. 6 otherwise is a joint pdf. k(xy?) 0<x<1, 0<y<1. is a joint pdf. Determine k, so that fx.x(x,y)= 1 otherwise 17. Determine k, so that fx.y(x,y)= kr 0<x<y<1 O otherwise is a joint pdf. k(x + y) 0<x< y<1 18. Determine k, so that fx. (x,y)= 1 0 otherwise is...
6. Justify briefly (max two lines) if the following statements are true or false. Answers with just Tor F won't receive any points. (a) (1 point) The curve with vector (Cartesian) equation r(t) = (t cost, tsint, t) lies on the surface with cylindrical equation 2 = 72 (b) (1 point) If ñ x p=0 then ñ = 0 or p=0. (c) (1 point) 24 – 4y2 lim (x,y)+(0,0) 22 + 2y2 = 0 (d) (1 point) If f(x, y)...
3. (a) The bus 500 arrives at Liverpool Airport at a rate of A buses per hour. Assume that the arrivals form a Poisson process. Let X (t) be the number of buses that arrive in t hours. X(t) is distributed as Px(o(u)=e-Ar (Xt)" u! when u is a positive integer and 0 otherwise. Let Y be the amount of time that you must wait for the 3rd bus to arrive. The event X (t) < 3 (fewer than three...
Mark each statement as True or False and justify your answer. a) The columns of a matrix A are linearly independent, if the equation Ax = 0 has the trivial solution. b) If vi, i = 1, ...,5, are in RS and V3 = 0, then {V1, V2, V3, V4, Vs} is linearly dependent. c) If vi, i = 1, 2, 3, are in R3, and if v3 is not a linear combination of vi and v2, then {V1, V2,...
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