Often relatively simple classes of pdfs are used if f

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continuous elements. Often relatively simple classes of PDFs are used. If F X x is specified for a continuous random vector, a PDF (for our purposes) is obtained by partial differentiation (except possibly at a small number of points): f X x F X x x 1 x 2 x m 8
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Suppose we have only two random variables, say X and Y . We can denote their joint PDF as f X , Y . Each has a marginal PDF , too: f X and f Y . Given the joint PDF it is easy to obtain the marginal PDFs (but the reverse is not true in general). If X , Y is discrete, taking values, say x 1 , x 2 ,... and y 1 , y 2 respectively, then f X x j 1 f X , Y x , y j j 1 P X x , Y y j f Y y j 1 f X , Y x j , y j 1 P X x j , Y y 9
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Of course, these expressions are valid in the case where one or both RVs takes on only a finite number of values. The infinite sums become finite sums. Sometimes one computes a joint distribution from an underlying experiment. 10
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EXAMPLE : Consider rolling two fair dice. Let X be the sum of the two dice and Y be largest of the two dice (which is the common value in the case of a tie). We know that there are 36 possible outcomes in the underlying sample space. X can take any value in 2,3,. ..,12 and Y can take any value in 1,2,. ..,6 , but the outcome on X is restricted by the outcome on Y . If D 1 is the number showing on the first die and D 2 that on the second, we can write X D 1 D 2 and Y max D 1 , D 2 . We can summarize the joint PDF of X , Y in a table. 11
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x 234567891 0 1 1 1 2 f Y y 1 1 36 0000000000 1 36 20 2 36 1 36 00000000 3 36 y 30 0 2 36 2 36 1 36 000000 5 36 4 000 2 36 2 36 2 36 1 36 0000 7 36 5 2 36 2 36 2 36 2 36 1 36 00 9 36 6 00000 2 36 2 36 2 36 2 36 2 36 1 36 11 36 f X x 1 36 2 36 3 36 4 36 5 36 6 36 5 36 4 36 3 36 2 36 1 36 12
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You should be able to verify that the final column gives the marginal PDF of Y (which you can obtain by direct reasoning) and that the final row gives the marginal PDF of X (also obtainable by just looking at the definition of X ). In the continuous case, we “integrate” out the other argument to get the marginal PDF. So f X x − f X , Y x , y dy − f X , Y x , v dv (just to emphasize that y in the first integral is just the argument of integration). 13
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EXAMPLE :Let X , Y be continuous random variables jointly distributed on the unit square as f X , Y x , y 6 xy 2 ,0 x 1, 0 y 1 (and zero elsewhere, as always). We can easily show this function integrates to one over the unit square by first integrating with respect to x , and then with respect to y (or vice versa): 0 1 0 1 f X , Y x , y dxdy 0 1 0 1 6 xy 2 dxdy 0 1 0 1 6 xdx y 2 dy 0 1 3 x 2 | 0 1 y 2 dy 0 1 3 y 2 dy y 3 | 0 1 1.
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Often relatively simple classes of PDFs are used If F X x...

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