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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 . Thus, the induced distribution is determined completely by the joint distribution function.
Distribution function for a discrete random vector
The induced distribution consists of point masses. At point in the range of there is probability mass . As in the general case, to determine 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.
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 on the plane. The region is a giant “sheet” with corner at . The value of 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 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 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 . The situation in the other cells may be checked out by this procedure.
Distribution function for a mixed distribution
The pair 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 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
Thus
This may be interpreted with the aid of [link] . Consider the sheet for point .
If we push the point up vertically, the upper boundary of is pushed up until eventually all probability mass on or to the left of the vertical line through is included. This is the total probability that . Now describes probability mass on the line. The probability mass described by is the same as the total joint probability mass on or to the left of the vertical line through . 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 .
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.
Thus, all the probability mass on the vertical line through is projected onto the point t i on a horizontal line to give . Similarly, all the probability mass on a horizontal line through is projected onto the point u j on a vertical line to give .
The pair produces a joint distribution that places mass 2/10 at each of the five points
(See [link] )
The marginal distribution for X has masses 2/10, 2/10, 4/10, 2/10 at points , respectively. Similarly, the marginal distribution for Y has masses 4/10, 4/10, 2/10 at points , respectively.
Consider again the joint distribution in [link] . The pair 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 , 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 in the amount of 0.2 produced by the two point masses on the vertical line. At , the total mass is “covered” and is constant at one for . By symmetry, the marginal distribution for Y is the same.
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