## Lagrange Multipliers Illinois Institute of Technology

STAT 400 Joint Probability Distributions. 2. Let X and Y be two independent random variables, each with the uni-form distribution on (0;1). Let M = min(X;Y) be the smaller of the two. a) Represent the event M > x as a region in the plane, and nd the, STAT 400 Joint Probability Distributions Fall 2017 1. Let X and Y have the joint p.d.f. f X, Y Find the value of C that would make f x, a valid probability density function. y b) Find the marginal probability density function of X, f X (x). c) Find the marginal probability density function of Y, f.

### Statistics 330 Assignment 3

self study Finding the PDF of Y where Y = min X - Cross. First of all, you want to be a bit more clear. The shape is revolved about the line x = 4. The boundaries of the shape being revolved are y = x 3, y = 0, and x = 2.. If you draw a diagram, you should see that beween using disks or cylindrical shells, the shells are the best choice., Find the extreme values of f(x,y)=x 2+2y2 on the disk x2+y в‰¤1. вЂўSolution: Compare the values of f at the critical points with values at the points on the boundary. Since f x =2x and f y =4y, the only critical point is (0,0). We compare the value of f at that point with the extreme values on the boundary from Example 2: вЂўf(0,0)=0 вЂўf(В±1,0)=1.

11/12/2008В В· Upload failed. Please upload a file larger than 100x100 pixels; We are experiencing some problems, please try again. You can only upload files of type PNG, JPG, or JPEG. 2. Let X and Y be two independent random variables, each with the uni-form distribution on (0;1). Let M = min(X;Y) be the smaller of the two. a) Represent the event M > x as a region in the plane, and nd the

3. (4 pts.) Suppose a certain test is to be taken by 3 students independently, and the time required by any student to complete the test has an exponential distribution with mean 1 Transforming a Random Variable Our purpose is to show how to find the density function fY of the transformation Y = g(X) of a random variable X with density function fX. Let X have probability density function (PDF) fX(x) and let Y = g(X). We want to find the PDF fY(y) of the random variable Y.

(a) Find the pdf of W = X2. (Noting that X has positive support.) (Noting that X has positive support.) w = x 2 which is one-to-one transformation as X is non-negative. Here, we will define jointly continuous random variables. Basically, two random variables are jointly continuous if they have a joint probability density function as defined below.

STAT 400 Joint Probability Distributions Fall 2017 1. Let X and Y have the joint p.d.f. f X, Y Find the value of C that would make f x, a valid probability density function. y b) Find the marginal probability density function of X, f X (x). c) Find the marginal probability density function of Y, f 7. Twoindependentuniformrandomvariables. Let X and Y be independently and uniformly drawn from the interval [0,1]. (a) Find the pdf of U = max(X,Y).

Example 5.1 Show that cosct and sinct are solutions of the second order ODE ВЁu +c2u = 0, where c is a constant. Deduce that Acosct+Bsinct is also a solution for arbitrary constants A,B. Remark It is conventional to use Л™u to denote the derivative (of u) with respect to t and ВЁu the second derivative with respect to t. In a similar way we will use u0 and u00 to denotes derivatives with 10/10/2013В В· This is from a chapter on distributions of two random variables. Let X and Y have the pdf f(x,y) = 1, 0

If we focus on the movement of X1, then the two lines overlap, only if 4 < X1 < 14. Now, given that 4 < X1 < 14, the two lines overlap if X1 < X2 +2 and X1 > X2 ВЎ2.That is, the probability of overlap is P(4 < X2 ВЎ2 < X1 < X2 +2 < 14): Since the two random variables are independent, the probability can be evaluated as P(4 < X2 ВЎ2 < X1 < X2 +2 < 14) = Z 16 6 ВµZ x 2+2 x2ВЎ2 1 4/26/2014В В· This answer assumes that X ~ N(0,1) and one needs to find pdf, E and V of Y=|X| Aliter:

### MTH135/STA104 Probability

HW6 DistributionsofFunctionsofRandomVariables. x 2+y = 25 and y(4xВЎ3y) = 0 The second of these equation s implies either that y = 0 or that 4 x = 3 y and both of these possibilities now need to be considered., Theorem IfX в€ј N(Вµ, Пѓ2)thentherandomvariableY = Xв€’Вµ Пѓ в€ј N(0, 1). Proof Let the random variable X have the normal distribution with probability density function fX(x)= 1 в€љ 2ПЂПѓ.

Find the volume of the solid revolved around y=x^3 y=0 x. 4.2 Conditional Distributions and Independence Deп¬Ѓnition 4.2.1 Let (X,Y) be a discrete bivariate random vector with joint pmf f(x,y) and Let (X,Y) be a continuous bivariate random vector with joint pdf f(x,y) and marginal pdfs fX(x) and fY (y). For any x such that fX(x) > 0, the conditional pdf of Y given that X = x, 2. Let X and Y be two independent random variables, each with the uni-form distribution on (0;1). Let M = min(X;Y) be the smaller of the two. a) Represent the event M > x as a region in the plane, and nd the.

### Find the probability density function of $Y=X^2

11 вЂ” TRANSFORMING DENSITY FUNCTIONS. The support of the joint pdf of X and Y is S= [0,1]2. The transformation The transformation u : ( x,y ) 7в†’( x/y,x + y ) maps S in the xy -plane into the domain u ( S ) in (a) Find the joint PDF of Xand Y. (b) Find the marginal PDF of Y. (c) Find the conditional PDF of Xgiven Y. (d) Find E[XjY = y], and use the total expectation theorem to nd E[X] in terms of E[Y]. (e) Use the symmetry of the problem to nd the value of E[X]. 2. Suppose X 1, X 1, and X 1 are independent exponential random variables, each with.

Find Derivative of y = x x. A calculus tutorial on how to find the first derivative of y = x x for x > 0. Note that the function defined by y = x x is neither a power function of the form x k nor an exponential function of the form b x and the formulas of Differentiation of these functions cannot be used. We need to find another method to find fX(x)fY (xв€’ z)dx. b) Find the pdf of Z := XY. Solution: Let us see how Method A will work out. Introduce a new pair of one-to-one RVs (Z = XY W = X. Solve it with respect to the old variables (Y = Z/W X = W. The corresponding Jacobian is J = 0 1

(a) Find the joint probability density function (pdf) of X,Y. Solution: Since they are independent it is just the product of a gamma density for X and a gamma density for Y. For the gamma distribution, Вµ = w/О», Пѓ2 = w/О»2. Since the mean and variance are both 3, О» = 1 and w = вЂ¦ Find Derivative of y = x x. A calculus tutorial on how to find the first derivative of y = x x for x > 0. Note that the function defined by y = x x is neither a power function of the form x k nor an exponential function of the form b x and the formulas of Differentiation of these functions cannot be used. We need to find another method to find

Lecture 16: Expected value, variance, independence and Chebyshev inequality Expected value, variance, and Chebyshev inequality. If Xis a random variable recall that the expected value of X, E[X] is the average value of X Expected value of X : E[X] = X P(X= ) 10/20/2015В В· In general, you are dealing with a function of two random variables. So given a joint PDF of [math] X,\,\,Y [/math], [math] f_{X,Y}(x, y) [/math], you can find the CDF of [math] Z [/math], [math] F_Z(z) [/math], by two variable integration. If a n...

(a) Find the pdf of W = X2. (Noting that X has positive support.) (Noting that X has positive support.) w = x 2 which is one-to-one transformation as X is non-negative. 11/12/2008В В· Upload failed. Please upload a file larger than 100x100 pixels; We are experiencing some problems, please try again. You can only upload files of type PNG, JPG, or JPEG.

First of all, you want to be a bit more clear. The shape is revolved about the line x = 4. The boundaries of the shape being revolved are y = x 3, y = 0, and x = 2.. If you draw a diagram, you should see that beween using disks or cylindrical shells, the shells are the best choice. (a) Find the joint probability density function (pdf) of X,Y. Solution: Since they are independent it is just the product of a gamma density for X and a gamma density for Y. For the gamma distribution, Вµ = w/О», Пѓ2 = w/О»2. Since the mean and variance are both 3, О» = 1 and w = вЂ¦

Example 5.1 Show that cosct and sinct are solutions of the second order ODE ВЁu +c2u = 0, where c is a constant. Deduce that Acosct+Bsinct is also a solution for arbitrary constants A,B. Remark It is conventional to use Л™u to denote the derivative (of u) with respect to t and ВЁu the second derivative with respect to t. In a similar way we will use u0 and u00 to denotes derivatives with 11/27/2011В В· Suppose X and Y are independent probability distributions with probability density functions f X (x) and f Y (y), and cumulative probability function F X (x) and F Y (y). If U=X+Y, then Given marginal pdfs of X and Y, find pdf of Z=X-Y Finding the PDF and CDF of вЂ¦

5/7/2016В В· How do you find the derivative of #[x+(x+sin^2x)^3]^4#? Calculus Differentiating Trigonometric Functions Intuitive Approach to the derivative of y=sin(x) 1 Answer 4.2 Conditional Distributions and Independence Deп¬Ѓnition 4.2.1 Let (X,Y) be a discrete bivariate random vector with joint pmf f(x,y) and Let (X,Y) be a continuous bivariate random vector with joint pdf f(x,y) and marginal pdfs fX(x) and fY (y). For any x such that fX(x) > 0, the conditional pdf of Y given that X = x

The known probability density function of X is, If then find the cumulative distribution function of Y. Now, the cumulative density function of, for and. HereвЂ™s the plot of is, Comment(0) Chapter , Problem is solved. View this answer. View a sample solution. View a full sample. Back to top. First of all, you want to be a bit more clear. The shape is revolved about the line x = 4. The boundaries of the shape being revolved are y = x 3, y = 0, and x = 2.. If you draw a diagram, you should see that beween using disks or cylindrical shells, the shells are the best choice.

## Stat 515 Final Exam Practice Problems + Solution

How do you find the derivative of [x+(x+sin^2x)^3]^4. Find the extreme values of f(x,y)=x 2+2y2 on the disk x2+y в‰¤1. вЂўSolution: Compare the values of f at the critical points with values at the points on the boundary. Since f x =2x and f y =4y, the only critical point is (0,0). We compare the value of f at that point with the extreme values on the boundary from Example 2: вЂўf(0,0)=0 вЂўf(В±1,0)=1, Lagrange multiplier examples Math 200-202 March 18, 2010 Example 1. Find the maximum and minimum values of the function f(x;y;z) = x2+y 2+z subject to the constraint x4+y4+z4 = 1. (Exercise #11 in Stewart,.

### Statistics 330 Assignment 3

AM 1650 Midterm Exam 2 Applied mathematics. The known probability density function of X is, If then find the cumulative distribution function of Y. Now, the cumulative density function of, for and. HereвЂ™s the plot of is, Comment(0) Chapter , Problem is solved. View this answer. View a sample solution. View a full sample. Back to top., 36-705 Brief Review of Basic Probability I assume you already know basic probability. Chapters 1-3 are a review. I will assume you have read and understood Chapters 1-3. If not, you should be in 36-700. 1 Random Variables Let be a sample space (a set of possible outcomes) with a probability distribution (also called a probability measure) P..

Find the pdf of Y when pdf of X is given. Ask Question Asked 6 years ago. Active 6 years ago. Viewed 1k times 3. 2 $\begingroup$ Find the pdf of Y. 2. Struggling with this PDF. 4. Conditional Expectation of pdf. 0. Computing the probability density function. Hot Network Questions 10/10/2013В В· This is from a chapter on distributions of two random variables. Let X and Y have the pdf f(x,y) = 1, 0

Lagrange multiplier examples Math 200-202 March 18, 2010 Example 1. Find the maximum and minimum values of the function f(x;y;z) = x2+y 2+z subject to the constraint x4+y4+z4 = 1. (Exercise #11 in Stewart, 4.2 Conditional Distributions and Independence Deп¬Ѓnition 4.2.1 Let (X,Y) be a discrete bivariate random vector with joint pmf f(x,y) and Let (X,Y) be a continuous bivariate random vector with joint pdf f(x,y) and marginal pdfs fX(x) and fY (y). For any x such that fX(x) > 0, the conditional pdf of Y given that X = x

x 2+y = 25 and y(4xВЎ3y) = 0 The second of these equation s implies either that y = 0 or that 4 x = 3 y and both of these possibilities now need to be considered. Find the extreme values of f(x,y)=x 2+2y2 on the disk x2+y в‰¤1. вЂўSolution: Compare the values of f at the critical points with values at the points on the boundary. Since f x =2x and f y =4y, the only critical point is (0,0). We compare the value of f at that point with the extreme values on the boundary from Example 2: вЂўf(0,0)=0 вЂўf(В±1,0)=1

The support of the joint pdf of X and Y is S= [0,1]2. The transformation The transformation u : ( x,y ) 7в†’( x/y,x + y ) maps S in the xy -plane into the domain u ( S ) in Note: General Form always has x 2 + y 2 for the first two terms. Going From General Form to Standard Form. Now imagine we have an equation in General Form: x 2 + y 2 + Ax + By + C = 0. How can we get it into Standard Form like this? (x-a) 2 + (y-b) 2 = r 2.

Theorem IfX в€ј N(Вµ, Пѓ2)thentherandomvariableY = Xв€’Вµ Пѓ в€ј N(0, 1). Proof Let the random variable X have the normal distribution with probability density function fX(x)= 1 в€љ 2ПЂПѓ 4/20/2017В В· Socratic Meta Featured Answers Topics How do you find the x and y intercepts of #2x-y=10#? Algebra Graphs of Linear Equations and Functions Intercepts by Substitution. 1 Answer Catherine T. Mohs В· MathFact-orials

STAT 400 Joint Probability Distributions Fall 2017 1. Let X and Y have the joint p.d.f. f X, Y Find the value of C that would make f x, a valid probability density function. y b) Find the marginal probability density function of X, f X (x). c) Find the marginal probability density function of Y, f 2. Let X and Y be two independent random variables, each with the uni-form distribution on (0;1). Let M = min(X;Y) be the smaller of the two. a) Represent the event M > x as a region in the plane, and nd the

10/10/2013В В· This is from a chapter on distributions of two random variables. Let X and Y have the pdf f(x,y) = 1, 0

4/26/2014В В· This answer assumes that X ~ N(0,1) and one needs to find pdf, E and V of Y=|X| Aliter: 10/20/2015В В· In general, you are dealing with a function of two random variables. So given a joint PDF of [math] X,\,\,Y [/math], [math] f_{X,Y}(x, y) [/math], you can find the CDF of [math] Z [/math], [math] F_Z(z) [/math], by two variable integration. If a n...

Find Derivative of y = x x. A calculus tutorial on how to find the first derivative of y = x x for x > 0. Note that the function defined by y = x x is neither a power function of the form x k nor an exponential function of the form b x and the formulas of Differentiation of these functions cannot be used. We need to find another method to find Here, we will define jointly continuous random variables. Basically, two random variables are jointly continuous if they have a joint probability density function as defined below.

Find the extreme values of f(x,y)=x 2+2y2 on the disk x2+y в‰¤1. вЂўSolution: Compare the values of f at the critical points with values at the points on the boundary. Since f x =2x and f y =4y, the only critical point is (0,0). We compare the value of f at that point with the extreme values on the boundary from Example 2: вЂўf(0,0)=0 вЂўf(В±1,0)=1 8. One Function of Two Random Variables Given two random variables X and Y and a function g(x,y), we form a new random variable Z as Given the joint p.d.f how does one obtain the p.d.f of Z ? Problems of this type are of interest from a practical standpoint. For example, a receiver output signal

4/26/2014В В· This answer assumes that X ~ N(0,1) and one needs to find pdf, E and V of Y=|X| Aliter: STAT 400 Joint Probability Distributions Fall 2017 1. Let X and Y have the joint p.d.f. f X, Y Find the value of C that would make f x, a valid probability density function. y b) Find the marginal probability density function of X, f X (x). c) Find the marginal probability density function of Y, f

Massachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Spring 2006) Problem 2: (25 points) Points breakdown: (a) 7 points; (b)вЂ“(d) 6 points each The continuous random variables X and Y have a joint pdf given by y x 2 1 1 2 11/12/2008В В· Upload failed. Please upload a file larger than 100x100 pixels; We are experiencing some problems, please try again. You can only upload files of type PNG, JPG, or JPEG.

Example 5.1 Show that cosct and sinct are solutions of the second order ODE ВЁu +c2u = 0, where c is a constant. Deduce that Acosct+Bsinct is also a solution for arbitrary constants A,B. Remark It is conventional to use Л™u to denote the derivative (of u) with respect to t and ВЁu the second derivative with respect to t. In a similar way we will use u0 and u00 to denotes derivatives with Lecture 16: Expected value, variance, independence and Chebyshev inequality Expected value, variance, and Chebyshev inequality. If Xis a random variable recall that the expected value of X, E[X] is the average value of X Expected value of X : E[X] = X P(X= )

Here, we will define jointly continuous random variables. Basically, two random variables are jointly continuous if they have a joint probability density function as defined below. 10/10/2013В В· This is from a chapter on distributions of two random variables. Let X and Y have the pdf f(x,y) = 1, 0

6. (1.7.22) Let Xhave the uniform pdf f X(x) = 1 Л‡ for Л‡ 2## Solved Let f(x) = в€’1 < x < 2 zero elsewhere be

Find the volume of the solid revolved around y=x^3 y=0 x. one way to proceed would be to substitute a new variable y for x2: Let y = x2 Replace the limits x = 0 and x = p ПЂ 2 by y = 0 and y = ПЂ 2 Replace 2xcosx2 by 2 в€љ y cosy Note that x = в€љ y and hence dx dy = 1 2 в€љ y and so replace dx by dy 2 в€љ y The original problem is thereby вЂ¦, Massachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Spring 2006) Problem 2: (25 points) Points breakdown: (a) 7 points; (b)вЂ“(d) 6 points each The continuous random variables X and Y have a joint pdf given by y x 2 1 1 2.

## Homework 9 (Math/Stats 425 Winter 2013) Statistics

self study Find the pdf of Y when pdf of X is given. 11/27/2011В В· Suppose X and Y are independent probability distributions with probability density functions f X (x) and f Y (y), and cumulative probability function F X (x) and F Y (y). If U=X+Y, then Given marginal pdfs of X and Y, find pdf of Z=X-Y Finding the PDF and CDF of вЂ¦ X() is the pdf of X which is given. Here are some more examples. Example 1 Suppose Xfollows the exponential distribution with = 1. If Y = p X nd the pdf of Y. Example 2 Let X ЛN(0;1). If Y = eX nd the pdf of Y. Note: Y it is said to have a log-normal distribution. Example 3 Let Xbe a continuous random variable with pdf f(x) = 2(1 x);0 x 1. If.

X() is the pdf of X which is given. Here are some more examples. Example 1 Suppose Xfollows the exponential distribution with = 1. If Y = p X nd the pdf of Y. Example 2 Let X ЛN(0;1). If Y = eX nd the pdf of Y. Note: Y it is said to have a log-normal distribution. Example 3 Let Xbe a continuous random variable with pdf f(x) = 2(1 x);0 x 1. If one way to proceed would be to substitute a new variable y for x2: Let y = x2 Replace the limits x = 0 and x = p ПЂ 2 by y = 0 and y = ПЂ 2 Replace 2xcosx2 by 2 в€љ y cosy Note that x = в€љ y and hence dx dy = 1 2 в€љ y and so replace dx by dy 2 в€љ y The original problem is thereby вЂ¦

5/7/2016В В· How do you find the derivative of #[x+(x+sin^2x)^3]^4#? Calculus Differentiating Trigonometric Functions Intuitive Approach to the derivative of y=sin(x) 1 Answer STAT 400 Joint Probability Distributions Fall 2017 1. Let X and Y have the joint p.d.f. f X, Y Find the value of C that would make f x, a valid probability density function. y b) Find the marginal probability density function of X, f X (x). c) Find the marginal probability density function of Y, f

fX(x)fY (xв€’ z)dx. b) Find the pdf of Z := XY. Solution: Let us see how Method A will work out. Introduce a new pair of one-to-one RVs (Z = XY W = X. Solve it with respect to the old variables (Y = Z/W X = W. The corresponding Jacobian is J = 0 1 STAT 400 Joint Probability Distributions Fall 2017 1. Let X and Y have the joint p.d.f. f X, Y Find the value of C that would make f x, a valid probability density function. y b) Find the marginal probability density function of X, f X (x). c) Find the marginal probability density function of Y, f

0.1 Recall: ordinary derivatives If y is a function of x then dy dx is the derivative meaning the gradient (slope of the graph) or the rate of change with respect to x. 0.2 Functions of 2 or more variables Note: General Form always has x 2 + y 2 for the first two terms. Going From General Form to Standard Form. Now imagine we have an equation in General Form: x 2 + y 2 + Ax + By + C = 0. How can we get it into Standard Form like this? (x-a) 2 + (y-b) 2 = r 2.

First of all, you want to be a bit more clear. The shape is revolved about the line x = 4. The boundaries of the shape being revolved are y = x 3, y = 0, and x = 2.. If you draw a diagram, you should see that beween using disks or cylindrical shells, the shells are the best choice. Example 5.1 Show that cosct and sinct are solutions of the second order ODE ВЁu +c2u = 0, where c is a constant. Deduce that Acosct+Bsinct is also a solution for arbitrary constants A,B. Remark It is conventional to use Л™u to denote the derivative (of u) with respect to t and ВЁu the second derivative with respect to t. In a similar way we will use u0 and u00 to denotes derivatives with

x 2+y = 25 and y(4xВЎ3y) = 0 The second of these equation s implies either that y = 0 or that 4 x = 3 y and both of these possibilities now need to be considered. 4.2 Conditional Distributions and Independence Deп¬Ѓnition 4.2.1 Let (X,Y) be a discrete bivariate random vector with joint pmf f(x,y) and Let (X,Y) be a continuous bivariate random vector with joint pdf f(x,y) and marginal pdfs fX(x) and fY (y). For any x such that fX(x) > 0, the conditional pdf of Y given that X = x

Massachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Spring 2006) Problem 2: (25 points) Points breakdown: (a) 7 points; (b)вЂ“(d) 6 points each The continuous random variables X and Y have a joint pdf given by y x 2 1 1 2 Please provide a detailed explanation. 1/rooty)if y <1 2/9 (-1 < = x < = 2) Find the pdf of Y = X2 The answer should be: f(y) =2/9(x+1) Get more help from Chegg Get 1:1 вЂ¦

Find the pdf of Y when pdf of X is given. Ask Question Asked 6 years ago. Active 6 years ago. Viewed 1k times 3. 2 $\begingroup$ Find the pdf of Y. 2. Struggling with this PDF. 4. Conditional Expectation of pdf. 0. Computing the probability density function. Hot Network Questions (a) Find the joint probability density function (pdf) of X,Y. Solution: Since they are independent it is just the product of a gamma density for X and a gamma density for Y. For the gamma distribution, Вµ = w/О», Пѓ2 = w/О»2. Since the mean and variance are both 3, О» = 1 and w = вЂ¦

Find the extreme values of f(x,y)=x 2+2y2 on the disk x2+y в‰¤1. вЂўSolution: Compare the values of f at the critical points with values at the points on the boundary. Since f x =2x and f y =4y, the only critical point is (0,0). We compare the value of f at that point with the extreme values on the boundary from Example 2: вЂўf(0,0)=0 вЂўf(В±1,0)=1 3. (4 pts.) Suppose a certain test is to be taken by 3 students independently, and the time required by any student to complete the test has an exponential distribution with mean 1

2. Let X and Y be two independent random variables, each with the uni-form distribution on (0;1). Let M = min(X;Y) be the smaller of the two. a) Represent the event M > x as a region in the plane, and nd the Find Derivative of y = x x. A calculus tutorial on how to find the first derivative of y = x x for x > 0. Note that the function defined by y = x x is neither a power function of the form x k nor an exponential function of the form b x and the formulas of Differentiation of these functions cannot be used. We need to find another method to find

Transforming a Random Variable Our purpose is to show how to find the density function fY of the transformation Y = g(X) of a random variable X with density function fX. Let X have probability density function (PDF) fX(x) and let Y = g(X). We want to find the PDF fY(y) of the random variable Y. (a) Find the joint probability density function (pdf) of X,Y. Solution: Since they are independent it is just the product of a gamma density for X and a gamma density for Y. For the gamma distribution, Вµ = w/О», Пѓ2 = w/О»2. Since the mean and variance are both 3, О» = 1 and w = вЂ¦

STAT 400 Joint Probability Distributions Fall 2017 1. Let X and Y have the joint p.d.f. f X, Y Find the value of C that would make f x, a valid probability density function. y b) Find the marginal probability density function of X, f X (x). c) Find the marginal probability density function of Y, f (a) Find the pdf of W = X2. (Noting that X has positive support.) (Noting that X has positive support.) w = x 2 which is one-to-one transformation as X is non-negative.

Transforming a Random Variable Our purpose is to show how to find the density function fY of the transformation Y = g(X) of a random variable X with density function fX. Let X have probability density function (PDF) fX(x) and let Y = g(X). We want to find the PDF fY(y) of the random variable Y. (b) Find the joint probability density function (pdf) for X,Y. Solution: We take the second order partial derivative of FX,Y (x,y) with respect to x and y. This gives fX,Y (x,y) = Л† 1 ПЂ 1 1+x2e в€’y, if y в‰Ґ 0 0, if y < 0 Note that X,Y are independent. X has the Cauchy distribution, and Y вЂ¦

X() is the pdf of X which is given. Here are some more examples. Example 1 Suppose Xfollows the exponential distribution with = 1. If Y = p X nd the pdf of Y. Example 2 Let X ЛN(0;1). If Y = eX nd the pdf of Y. Note: Y it is said to have a log-normal distribution. Example 3 Let Xbe a continuous random variable with pdf f(x) = 2(1 x);0 x 1. If x 2+y = 25 and y(4xВЎ3y) = 0 The second of these equation s implies either that y = 0 or that 4 x = 3 y and both of these possibilities now need to be considered.

2. Let X and Y be two independent random variables, each with the uni-form distribution on (0;1). Let M = min(X;Y) be the smaller of the two. a) Represent the event M > x as a region in the plane, and nd the Find the extreme values of f(x,y)=x 2+2y2 on the disk x2+y в‰¤1. вЂўSolution: Compare the values of f at the critical points with values at the points on the boundary. Since f x =2x and f y =4y, the only critical point is (0,0). We compare the value of f at that point with the extreme values on the boundary from Example 2: вЂўf(0,0)=0 вЂўf(В±1,0)=1

7. Twoindependentuniformrandomvariables. Let X and Y be independently and uniformly drawn from the interval [0,1]. (a) Find the pdf of U = max(X,Y). 8. One Function of Two Random Variables Given two random variables X and Y and a function g(x,y), we form a new random variable Z as Given the joint p.d.f how does one obtain the p.d.f of Z ? Problems of this type are of interest from a practical standpoint. For example, a receiver output signal

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