4.4-JG3 The arrival location, x, and arrival altitude, y, of airplanes is randomly distributed on either...
4.4-JG3
The arrival location, x, and arrival altitude, y, of airplanes is
randomly distributed on either side of a city with
fx,y(x,y)=(25/(23ab))(y/a)[1-(x/b)^4(y/a)^3] for -b<x<b and
0<y<a and zero elsewhere. Determine fy(y|-b<X<0). Carry
a and b as constants to simplify the math. Ans:
(50/(23a))(y/a)[1-(1/5)(y/a)^3]
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4.4-JG3
The arrival location, x, and arrival altitude, y, of airplanes is
randomly distributed on either...
A random variable Y is a function of random variable X, where y=x^3 and fx(x)=1 from...
A random variable Y is a function of random variable X, where y=x^3 and fx(x)=1 from 0 to 1 and =0 elsewhere. Determine fy(y). Ans: fy(y)=(1/3)y^(-2/3) for 0<y<1
question with answers, show steps: Given: fx,y(x,y)= (5/16)yx^2 for 0<y<x<2, determine: a) fx(x) by integrating y...
question with answers, show steps: Given: fx,y(x,y)= (5/16)yx^2 for 0<y<x<2, determine: a) fx(x) by integrating y from 0 to x. Ans: fx(x)=(5/32)x^4 for 0<x<2 b) fy(y) by integrating x from y to 2 Ans: fy(y)=(5/48)y(8-y^3) for y<x<2 c) Test for independence using Criterion b Ans: Fails-> Not independent
Suppose that X is uniformly distributed between 0 and 1. Given X = x, Y is...
Suppose that X is uniformly distributed between 0 and 1. Given X = x, Y is uniformly distributed between 0 and x2. (a) Determine E(Y |X = x) and then Var(Y |X = x). Is E(Y |X = x) a linear function of x? (b) Find f(x, y) using fX(x) and fY |X(y|x). (c) Find fY (y). (d) Find the conditional density of X given Y = y. (e) Find the correlation coefficient between X and Y .
1. Consider a continuous random variable X with the probability density function Sx(x) = 3<x<7, zero...
1. Consider a continuous random variable X with the probability density function Sx(x) = 3<x<7, zero elsewhere. a) Find the value of C that makes fx(x) a valid probability density function. b) Find the cumulative distribution function of X, Fx(x). "Hint”: To double-check your answer: should be Fx(3)=0, Fx(7)=1. 1. con (continued) Consider Y=g(x)- 20 100 X 2 + Find the support (the range of possible values) of the probability distribution of Y. d) Use part (b) and the c.d.f....
) Let X, Y be two random variables with the following properties. Y had density function...
) Let X, Y be two random variables with the following
properties. Y had
density function fY (y) = 3y
2
for 0 < y < 1 and zero elsewhere. For 0 < y < 1, given
Y = y, X
had conditional density function fX|Y (x | y) = 2x
y
2 for 0 < x < y and zero elsewhere.
(a) Find the joint density function fX,Y . Be precise about where
the values (x, y) are non-zero....
Suppose y has a「(1,1) distribution while X given y has the conditional pdf elsewhere 0 Note that both the pdf of Y and the conditional pdf are easy to simulate. (a) Set up the following algorithm to...
Suppose y has a「(1,1) distribution while X given y has the conditional pdf elsewhere 0 Note that both the pdf of Y and the conditional pdf are easy to simulate. (a) Set up the following algorithm to generate a stream of iid observations with pdf fx(x) 1. Generate y ~ fy(y). 2. Generate X~fxy(XY), (b) How would you estimate E[X]?
Suppose y has a「(1,1) distribution while X given y has the conditional pdf elsewhere 0 Note that both the pdf...
(II) Multiple continuous random variables: 8.2 Let X and Y have joint density fXY(x,y) = cx^2y...
(II) Multiple continuous random variables: 8.2 Let X and Y have joint density fXY(x,y) = cx^2y for x and y in the triangle defined by 0 < x < 1, 0 < y < 1, 0 < x + y < 1 and fXY(x,y) = 0 elsewhere. a. What is c? b. What are the marginals fX(x) and fY(y)? c. What are E[X], E[Y], Var[X] and Var[Y]? d. What is E[XY]? Are X and Y independent?
A point (X, Y ) in the Cartesian plane is uniformly distributed within the unit circle if X and Y have joint density Fi...
A point (X, Y ) in the Cartesian plane is uniformly distributed
within the unit circle if X and Y have joint density
Find the marginal densities fX and fY and state whether
X and Y are independent or not. Provide a mathematical
justification for your answer.
1, 22 + y2 <1, f(x, y) = { 1 0, otherwise.
Problem 1 Suppose X ~ fx(x), and let Y = aX + b. We know that...
Problem 1 Suppose X ~ fx(x), and let Y = aX + b. We know that E(Y) = aE(X) b, and Var(X)a2Var(X). What about the density of Y, fy(y)? Assuming a > 0. Calculate fy(y) using the following two methods (1) Let Fx() P(X x). Calculate Fy(y) = P(Y < y) in terms of Fx. Then calculate fy (2) Calculate Y (y, y + Ay)) Ay fr(y) (3) Give geometric explanations of your result
Need help with question 2 (not question 1) 1. Suppose that (X,Y) is uniformly distributed over...
Need help with question 2 (not question
1)
1. Suppose that (X,Y) is uniformly distributed over the region {(x, y): 0 < \y< x < 1}. Find: a) the joint density of (X, Y); b) the marginal densities fx(x) and fy(y). c) Are X and Y independent? d) Find E(X) and E(Y). 2. Repeat Exercise 1 for (X,Y) with uniform distribution over {(x, y): 0 < \x]+\y< 1}.
1. Consider a continuous random variable X with the probability density function Sx(x) = 3<x<7, zero elsewhere. a) Find the value of C that makes fx(x) a valid probability density function. b) Find the cumulative distribution function of X, Fx(x). "Hint”: To double-check your answer: should be Fx(3)=0, Fx(7)=1. 1. con (continued) Consider Y=g(x)- 20 100 X 2 + Find the support (the range of possible values) of the probability distribution of Y. d) Use part (b) and the c.d.f....
) Let X, Y be two random variables with the following
properties. Y had
density function fY (y) = 3y
2
for 0 < y < 1 and zero elsewhere. For 0 < y < 1, given
Y = y, X
had conditional density function fX|Y (x | y) = 2x
y
2 for 0 < x < y and zero elsewhere.
(a) Find the joint density function fX,Y . Be precise about where
the values (x, y) are non-zero....
Suppose y has a「(1,1) distribution while X given y has the conditional pdf elsewhere 0 Note that both the pdf of Y and the conditional pdf are easy to simulate. (a) Set up the following algorithm to generate a stream of iid observations with pdf fx(x) 1. Generate y ~ fy(y). 2. Generate X~fxy(XY), (b) How would you estimate E[X]?
Suppose y has a「(1,1) distribution while X given y has the conditional pdf elsewhere 0 Note that both the pdf...
A point (X, Y ) in the Cartesian plane is uniformly distributed
within the unit circle if X and Y have joint density
Find the marginal densities fX and fY and state whether
X and Y are independent or not. Provide a mathematical
justification for your answer.
1, 22 + y2 <1, f(x, y) = { 1 0, otherwise.
Problem 1 Suppose X ~ fx(x), and let Y = aX + b. We know that E(Y) = aE(X) b, and Var(X)a2Var(X). What about the density of Y, fy(y)? Assuming a > 0. Calculate fy(y) using the following two methods (1) Let Fx() P(X x). Calculate Fy(y) = P(Y < y) in terms of Fx. Then calculate fy (2) Calculate Y (y, y + Ay)) Ay fr(y) (3) Give geometric explanations of your result
Need help with question 2 (not question
1)
1. Suppose that (X,Y) is uniformly distributed over the region {(x, y): 0 < \y< x < 1}. Find: a) the joint density of (X, Y); b) the marginal densities fx(x) and fy(y). c) Are X and Y independent? d) Find E(X) and E(Y). 2. Repeat Exercise 1 for (X,Y) with uniform distribution over {(x, y): 0 < \x]+\y< 1}.
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