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Joint absolutely continuous random variables

In the single-variable case, the condition that there are no point mass concentrations on the line ensures the existence of a probability density function, useful in probability calculations.A similar situation exists for a joint distribution for two (or more) variables. For any joint mapping to the plane which assigns zero probability to eachset with zero area (discrete points, line or curve segments, and countable unions of these) there is a density function.

Definition

If the joint probability distribution for the pair { X , Y } assigns zero probability to every set of points with zero area, then there exists a joint density function f X Y with the property

P [ ( X , Y ) Q ] = Q f X Y

We have three properties analogous to those for the single-variable case:

(f1) f X Y 0 (f2) R 2 f X Y = 1 (f3) F X Y ( t , u ) = - t - u f X Y

At every continuity point for f X Y , the density is the second partial

f X Y ( t , u ) = 2 F X Y ( t , u ) t u

Now

F X ( t ) = F X Y ( t , ) = - t - f X Y ( r , s ) d s d r

A similar expression holds for F Y ( u ) . Use of the fundamental theorem of calculus to obtain the derivatives gives the result

f X ( t ) = - f X Y ( t , s ) d s and f Y ( u ) = - f X Y ( r , u ) d u

Marginal densities . Thus, to obtain the marginal density for the first variable, integrate out the second variablein the joint density, and similarly for the marginal for the second variable.

Marginal density functions

Let f X Y ( t , u ) = 8 t u 0 u t 1 . This region is the triangle bounded by u = 0 , u = t , and t = 1 (see [link] )

f X ( t ) = f X Y ( t , u ) d u = 8 t 0 t u d u = 4 t 3 , 0 t 1
f Y ( u ) = f X Y ( t , u ) d t = 8 u u 1 t d t = 4 u ( 1 - u 2 ) , 0 u 1

P ( 0 . 5 X 0 . 75 , Y > 0 . 5 ) = P [ ( X , Y ) Q ] where Q is the common part of the triangle with the strip between t = 0 . 5 and t = 0 . 75 and above the line u = 0 . 5 . This is the small triangle bounded by u = 0 . 5 , u = t , and t = 0 . 75 . Thus

p = 8 1 / 2 3 / 4 1 / 2 t t u d u d t = 25 / 256 0 . 0977
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A graph with a line rising from point (0,0) to (1,1) and another line rising perpendicularly from point (1,0) forming a corner with the other line at point (1,1). The resulting triangle is contains the function f_xy(t,u)=8tu. Along the diagonal line at point (0.5,0.5) a line extends to the right to another point (0.5, 0.75). Line segments extend upward from both of these points and the resulting triangle is shaded with the letter Q in the middle. A graph with a line rising from point (0,0) to (1,1) and another line rising perpendicularly from point (1,0) forming a corner with the other line at point (1,1). The resulting triangle is contains the function f_xy(t,u)=8tu. Along the diagonal line at point (0.5,0.5) a line extends to the right to another point (0.5, 0.75). Line segments extend upward from both of these points and the resulting triangle is shaded with the letter Q in the middle.
Distribution for [link] .

Marginal distribution with compound expression

The pair { X , Y } has joint density f X Y ( t , u ) = 6 37 ( t + 2 u ) on the region bounded by t = 0 , t = 2 , u = 0 , and u = max { 1 , t } (see [link] ). Determine the marginal density f X .

SOLUTION

Examination of the figure shows that we have different limits for the integral with respect to u for 0 t 1 and for 1 < t 2 .

  • For 0 t 1
    f X ( t ) = 6 37 0 1 ( t + 2 u ) d u = 6 37 ( t + 1 )
  • For 1 < t 2
    f X ( t ) = 6 37 0 t ( t + 2 u ) d u = 12 37 t 2

We may combine these into a single expression in a manner used extensively in subsequent treatments. Suppose M = [ 0 , 1 ] and N = ( 1 , 2 ] . Then I M ( t ) = 1 for t M (i.e., 0 t 1 ) and zero elsewhere. Likewise, I N ( t ) = 1 for t N and zero elsewhere. We can, therefore express f X by

f X ( t ) = I M ( t ) 6 37 ( t + 1 ) + I N ( t ) 12 37 t 2
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A graph with a horizontal line extending from the point (0,1) to (1,1) and is labeled u=1. Then another line proceeds at a diagonal from (1,1) to (2,2) and is labeled u=t. Another line rises to this point, (2,2) from point (2,0) and is labeled t=2. this figure contains the equation f_xy(t,u)=(6/37)(t+2u). A graph with a horizontal line extending from the point (0,1) to (1,1) and is labeled u=1. Then another line proceeds at a diagonal from (1,1) to (2,2) and is labeled u=t. Another line rises to this point, (2,2) from point (2,0) and is labeled t=2. this figure contains the equation f_xy(t,u)=(6/37)(t+2u).
Marginal distribution for [link] .

Discrete approximation in the continuous case

For a pair { X , Y } with joint density f X Y , we approximate the distribution in a manner similar to that for a single random variable. We then utilize the techniquesdeveloped for a pair of simple random variables. If we have n approximating values t i for X and m approximating values u j for Y , we then have n · m pairs ( t i , u j ) , corresponding to points on the plane. If we subdivide the horizontal axis for values of X , with constant increments d x , as in the single-variable case, and the vertical axis for values of Y , with constant increments d y , we have a grid structure consisting of rectangles of size d x · d y . We select t i and u j at the midpoint of its increment, so that the point ( t i , u j ) is at the midpoint of the rectangle. If we let the approximating pair be { X * , Y * } , we assign

p i j = P ( X * , Y * ) = ( t i , u j ) = P ( X * = t i , Y * = u j ) = P ( ( X , Y ) in i j th rectangle)

As in the one-variable case, if the increments are small enough,

P ( ( X , Y ) i j th rectangle) d x · d y · f X Y ( t i , u j )

The m-procedure tuappr calls for endpoints of intervals which include the ranges of X and Y and for the numbers of subintervals on each. It then prompts for an expression for f X Y ( t , u ) , from which it determines the joint probability distribution. It calculates the marginal approximatedistributions and sets up the calculating matrices t and u as does the m-process jcalc for simple random variables. Calculations are then carried outas for any joint simple pair.

Approximation to a joint continuous distribution

f X Y ( t , u ) = 3 on 0 u t 2 1

Determine P ( X 0 . 8 , Y > 0 . 1 ) .

>> tuappr Enter matrix [a b] of X-range endpoints  [0 1] Enter matrix [c d] of Y-range endpoints  [0 1] Enter number of X approximation points  200Enter number of Y approximation points  200 Enter expression for joint density  3*(u <= t.^2) Use array operations on X, Y, PX, PY, t, u, and P>> M = (t <= 0.8)&(u > 0.1);>> p = total(M.*P)          % Evaluation of the integral with p =   0.3355                % Maple gives 0.3352455531
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The discrete approximation may be used to obtain approximate plots of marginal distribution and density functions.

A graph of marginal density and distribution for X. The line for fx extends from point (-1,0) to a sharp apex at (0,1.5 and then the line declines mirroring the previous side to a point (1,0). The line for FX is dashed and begins at (-0.8,0) and ascends gently and plateaus around (0.8,1). A graph of marginal density and distribution for X. The line for fx extends from point (-1,0) to a sharp apex at (0,1.5 and then the line declines mirroring the previous side to a point (1,0). The line for FX is dashed and begins at (-0.8,0) and ascends gently and plateaus around (0.8,1).
Marginal density and distribution function for [link] .

Approximate plots of marginal density and distribution functions

f X Y ( t , u ) = 3 u on the triangle bounded by u = 0 , u 1 + t , and u 1 - t .

>> tuappr Enter matrix [a b] of X-range endpoints  [-1 1] Enter matrix [c d] of Y-range endpoints  [0 1] Enter number of X approximation points  400Enter number of Y approximation points  200 Enter expression for joint density  3*u.*(u<=min(1+t,1-t)) Use array operations on X, Y, PX, PY, t, u, and P>> fx = PX/dx;                % Density for X  (see  [link] )                               % Theoretical (3/2)(1 - |t|)^2>> fy = PY/dy;                % Density for Y>> FX = cumsum(PX);           % Distribution function for X ( [link] )>> FY = cumsum(PY);           % Distribution function for Y>> plot(X,fx,X,FX)            % Plotting details omitted
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These approximation techniques useful in dealing with functions of random variables, expectations, and conditional expectation and regression.

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Source:  OpenStax, Applied probability. OpenStax CNX. Aug 31, 2009 Download for free at http://cnx.org/content/col10708/1.6
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