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The theoretical result quoted in the real variable case extends to ensure that a distribution on the plane is determined uniquely by consistent assignments to the semiinfinite intervals Q t u . Thus, the induced distribution is determined completely by the joint distribution function.

A diagram showing the ragion of a graph representing Q_ab. It is a shaded square plotted on a typical two dimension graph. A diagram showing the ragion of a graph representing Q_ab. It is a shaded square plotted on a typical two dimension graph.
The region Q a b for the value F X Y ( a , b ) .

Distribution function for a discrete random vector

The induced distribution consists of point masses. At point ( t i , u j ) in the range of W = ( X , Y ) there is probability mass p i j = P [ W = ( t i , u j ) ] = P ( X = t i , Y = u j ) . As in the general case, to determine P [ ( X , Y ) Q ] we determine how much probability mass is in the region. In the discrete case (or in any case where there are point massconcentrations) one must be careful to note whether or not the boundaries are included in the region, should there be mass concentrations on the boundary.

A graph of the joint distribution for Example 1. A graph of the joint distribution for Example 1.
The joint distribution for [link] .

Distribution function for the selection problem in [link]

The probability distribution is quite simple. Mass 3/10 at (0,2), 6/10 at (1,1), and 1/10 at (2,0). This distribution is plotted in [link] . To determine (and visualize) the joint distribution function, think of moving the point ( t , u ) on the plane. The region Q t u is a giant “sheet” with corner at ( t , u ) . The value of F X Y ( t , u ) is the amount of probability covered by the sheet. This value is constant over any grid cell, including the left-hand and lower boundariies, and is the value taken on at the lowerleft-hand corner of the cell. Thus, if ( t , u ) is in any of the three squares on the lower left hand part of the diagram, no probability mass is coveredby the sheet with corner in the cell. If ( t , u ) is on or in the square having probability 6/10 at the lower left-hand corner, then the sheet covers that probability, and the value of F X Y ( t , u ) = 6 / 10 . The situation in the other cells may be checked out by this procedure.

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Distribution function for a mixed distribution

A mixed distribution

The pair { X , Y } produces a mixed distribution as follows (see [link] )

Point masses 1/10 at points (0,0), (1,0), (1,1), (0,1)

Mass 6/10 spread uniformly over the unit square with these vertices

The joint distribution function is zero in the second, third, and fourth quadrants.

  • If the point ( t , u ) is in the square or on the left and lower boundaries, the sheet covers the point mass at (0,0) plus 0.6 times the area covered within the square.Thus in this region
    F X Y ( t , u ) = 1 10 ( 1 + 6 t u )
  • If the pont ( t , u ) is above the square (including its upper boundary) but to the left of the line t = 1 , the sheet covers two point masses plus the portion of the mass in the square to the left of the vertical line through ( t , u ) . In this case
    F X Y ( t , u ) = 1 10 ( 2 + 6 t )
  • If the point ( t , u ) is to the right of the square (including its boundary) with 0 u < 1 , the sheet covers two point masses and the portion of the mass in the square below the horizontal line through ( t , u ) , to give
    F X Y ( t , u ) = 1 10 ( 2 + 6 u )
  • If ( t , u ) is above and to the right of the square (i.e., both 1 t and 1 u ). then all probability mass is covered and F X Y ( t , u ) = 1 in this region.
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The Mixed joint distribution for example 1. The graph has a shaded box formed by a two intersecting lines and the x and y axis. The intersection of the two line segments is labeled {l,u}. Further along the x and y axis two more line segments extend perpendicularly to their respective axis and intersect at (1,1). About this box is the phrase 'Point mass 1/10 al each vertex'. Inside the box created by the outer lines is the phrase 'Mass 6/10 spread uniformly on the square. Density 0.6'. Underneath the entire is the phrase 'Mass 0.1+0.6tu in region covered by infinite sheet with corner at (t,u). The Mixed joint distribution for example 1. The graph has a shaded box formed by a two intersecting lines and the x and y axis. The intersection of the two line segments is labeled {l,u}. Further along the x and y axis two more line segments extend perpendicularly to their respective axis and intersect at (1,1). About this box is the phrase 'Point mass 1/10 al each vertex'. Inside the box created by the outer lines is the phrase 'Mass 6/10 spread uniformly on the square. Density 0.6'. Underneath the entire is the phrase 'Mass 0.1+0.6tu in region covered by infinite sheet with corner at (t,u).
Mixed joint distribution for [link] .

Marginal distributions

If the joint distribution for a random vector is known, then the distribution for each of the component random variables may be determined. These are known as marginal distributions . In general, the converse is not true. However, if the component random variables form an independent pair, the treatment in that case shows that the marginals determine the jointdistribution.

To begin the investigation, note that

F X ( t ) = P ( X t ) = P ( X t , Y < ) i.e., Y can take any of its possible values

Thus

F X ( t ) = F X Y ( t , ) = lim u F X Y ( t , u )

This may be interpreted with the aid of [link] . Consider the sheet for point ( t , u ) .

A graph showing the construction for obtaining the marginal distribution for X. The x axis is labeled t and the y-axis is labeled u. This diagram is complex. Perpendicular to the x axis there are two line segments. The first line segment is a dotted line and then next to that is the second line which is a solid line. The area between these two lines is shaded gray. These lines and this shaded area extends on both side of the x-axis. Perpendicular to the y-axis, a dashed line extends on both sides of the y-axis and ends when it reaches the solid line ascending from the x-axis forming a box. Inside the resulting box is Q_tu, with an arrow pointing up and intersecting the dashed line. Above this dashed line is the phrase 'Half plane Q_t'. To the right of the solid vertical line and above where the horizontal dashed line ends there is the phrase 'Boundary moves up to include all probability mass in the half plane' with an arrow pointing down and to the left and ending at the horizontal dashed line. To the right of this dashed line and on the right side of the solid vertical line there is the phrase 'u increases without limit'. Below the x-axis and to the right of the solid vertical line is the phrase 'f_X(t)= probability in the half plane=F_XY(t,∞)'. A graph showing the construction for obtaining the marginal distribution for X. The x axis is labeled t and the y-axis is labeled u. This diagram is complex. Perpendicular to the x axis there are two line segments. The first line segment is a dotted line and then next to that is the second line which is a solid line. The area between these two lines is shaded gray. These lines and this shaded area extends on both side of the x-axis. Perpendicular to the y-axis, a dashed line extends on both sides of the y-axis and ends when it reaches the solid line ascending from the x-axis forming a box. Inside the resulting box is Q_tu, with an arrow pointing up and intersecting the dashed line. Above this dashed line is the phrase 'Half plane Q_t'. To the right of the solid vertical line and above where the horizontal dashed line ends there is the phrase 'Boundary moves up to include all probability mass in the half plane' with an arrow pointing down and to the left and ending at the horizontal dashed line. To the right of this dashed line and on the right side of the solid vertical line there is the phrase 'u increases without limit'. Below the x-axis and to the right of the solid vertical line is the phrase 'f_X(t)= probability in the half plane=F_XY(t,∞)'.
Construction for obtaining the marginal distribution for X .

If we push the point up vertically, the upper boundary of Q t u is pushed up until eventually all probability mass on or to the left of the vertical line through ( t , u ) is included. This is the total probability that X t . Now F X ( t ) describes probability mass on the line. The probability mass described by F X ( t ) is the same as the total joint probability mass on or to the left of the vertical line through ( t , u ) . We may think of the mass in the half plane being projected onto the horizontal line to give the marginal distribution for X . A parallel argument holds for the marginal for Y .

F Y ( u ) = P ( Y u ) = F X Y ( , u ) = mass on or below horizontal line through ( t , u )

This mass is projected onto the vertical axis to give the marginal distribution for Y .

Marginals for a joint discrete distribution

Consider a joint simple distribution.

P ( X = t i ) = j = 1 m P ( X = t i , Y = u j ) and P ( Y = u j ) = i = 1 n P ( X = t i , Y = u j )

Thus, all the probability mass on the vertical line through ( t i , 0 ) is projected onto the point t i on a horizontal line to give P ( X = t i ) . Similarly, all the probability mass on a horizontal line through ( 0 , u j ) is projected onto the point u j on a vertical line to give P ( Y = u j ) .

Marginals for a discrete distribution

The pair { X , Y } produces a joint distribution that places mass 2/10 at each of the five points

( 0 , 0 ) , ( 1 , 1 ) , ( 2 , 0 ) , ( 2 , 2 ) , ( 3 , 1 ) (See [link] )

The marginal distribution for X has masses 2/10, 2/10, 4/10, 2/10 at points t = 0 , 1 , 2 , 3 , respectively. Similarly, the marginal distribution for Y has masses 4/10, 4/10, 2/10 at points u = 0 , 1 , 2 , respectively.

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A graph showing the marginal distribution for X. The figure consist of a 4x3 grid of dashed lines creating on the positive side of a two dimensional graph. Below this element of the figure is the second element which consist of a line segment with four hollow circle situated on the line. A graph showing the marginal distribution for X. The figure consist of a 4x3 grid of dashed lines creating on the positive side of a two dimensional graph. Below this element of the figure is the second element which consist of a line segment with four hollow circle situated on the line.
Marginal distribution for Example 1.

Consider again the joint distribution in [link] . The pair { X , Y } produces a mixed distribution as follows:

Point masses 1/10 at points (0,0), (1,0), (1,1), (0,1)

Mass 6/10 spread uniformly over the unit square with these vertices

The construction in [link] shows the graph of the marginal distribution function F X . There is a jump in the amount of 0.2 at t = 0 , corresponding to the two point masses on the vertical line. Then the mass increases linearly with t , slope 0.6, until a final jump at t = 1 in the amount of 0.2 produced by the two point masses on the vertical line. At t = 1 , the total mass is “covered” and F X ( t ) is constant at one for t 1 . By symmetry, the marginal distribution for Y is the same.

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The figure is very complex. It consist of two separate elements situated vertically to one another. The upper figure is a two dimensional graph, a line segments extends from each of axes  and forms a square. Bisecting this square is another line. To the left of this line the square is shaded, while the an arrow points to the right side with the phrase 'Mass 6/10 spread uniformly on the square. Density 0.6. Above the square there is the phrase 'Point masses 1/10 at each vertex'. Below the square there is a squiggly arrow point to the shaded portion of the square with the phrase 'Mass 0.2+0.6t covered by the half plane.' Below this element is another graph. At 0.3 there is a line extending up and to the right with an arrow pointing to it labeled F_X(t)=0.2+0.6t. On the y-axis there two horizontal lines above the point at which the the previously discussed line segment begins. The first of these lines is labeled 0.8 the second is labeled 1. To the far right of the line labeled 1 there is line segment extending to the right. Below this figure is the label 'Marginal distribution for X'. The figure is very complex. It consist of two separate elements situated vertically to one another. The upper figure is a two dimensional graph, a line segments extends from each of axes  and forms a square. Bisecting this square is another line. To the left of this line the square is shaded, while the an arrow points to the right side with the phrase 'Mass 6/10 spread uniformly on the square. Density 0.6. Above the square there is the phrase 'Point masses 1/10 at each vertex'. Below the square there is a squiggly arrow point to the shaded portion of the square with the phrase 'Mass 0.2+0.6t covered by the half plane.' Below this element is another graph. At 0.3 there is a line extending up and to the right with an arrow pointing to it labeled F_X(t)=0.2+0.6t. On the y-axis there two horizontal lines above the point at which the the previously discussed line segment begins. The first of these lines is labeled 0.8 the second is labeled 1. To the far right of the line labeled 1 there is line segment extending to the right. Below this figure is the label 'Marginal distribution for X'.
Marginal distribution for [link] .

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
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Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
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Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
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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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