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i.e., the sum over all values of x and y is 1. Suppose that X can assume any one of m values x1, x2, . . . , xm and Y can assume any one of n values y1, y2, . . . , yn. Then the probability of the event that X xj and Y yk is given by P(X xj, Y yk) f(xj, yk) (14)
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A joint probability function for X and Y can be represented by a joint probability table as in Table 2-3. The probability that X xj is obtained by adding all entries in the row corresponding to xi and is given by
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Table 2-3 Y X x1 x2 ( xm Totals S y1 f (x1, y1) f (x2, y1) ( f (xm, y1 ) f2 (y1 ) y2 f (x1, y2) f (x2, y2) ( f (xm, y2 ) f2 (y2 ) c c c c c yn f(x1, yn ) f(x2, yn ) ( f(xm, yn) f2 (yn) Totals T f1 (x1) f1 (x2) ( f1 (xm) 1 d Grand Total
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For j 1, 2, . . . , m, these are indicated by the entry totals in the extreme right-hand column or margin of Table 2-3. Similarly the probability that Y yk is obtained by adding all entries in the column corresponding to yk and is given by
f2(yk )
a f (xj, yk )
(16)
For k 1, 2, . . . , n, these are indicated by the entry totals in the bottom row or margin of Table 2-3. Because the probabilities (15) and (16) are obtained from the margins of the table, we often refer to f1(xj) and f2(yk) [or simply f1(x) and f2(y)] as the marginal probability functions of X and Y, respectively.
CHAPTER 2 Random Variables and Probability Distributions
It should also be noted that
a f1 (xj)
1 a f2 (yk)
(17)
which can be written
a a f (xj, yk)
j 1k 1
(18)
This is simply the statement that the total probability of all entries is 1. The grand total of 1 is indicated in the lower right-hand corner of the table. The joint distribution function of X and Y is defined by F(x, y) P(X x, Y y) a a f (u, v) y. (19)
u xv y
In Table 2-3, F(x, y) is the sum of all entries for which xj
x and yk
2. CONTINUOUS CASE. The case where both variables are continuous is obtained easily by analogy with the discrete case on replacing sums by integrals. Thus the joint probability function for the random variables X and Y (or, as it is more commonly called, the joint density function of X and Y ) is defined by 1. f(x, y)
` ` `
2. 3
f (x, y) dx dy
Graphically z f(x, y) represents a surface, called the probability surface, as indicated in Fig. 2-4. The total volume bounded by this surface and the xy plane is equal to 1 in accordance with Property 2 above. The probability that X lies between a and b while Y lies between c and d is given graphically by the shaded volume of Fig. 2-4 and mathematically by
b, c
3 a y
f (x, y) dx dy
(20)
Fig. 2-4
More generally, if A represents any event, there will be a region 5A of the xy plane that corresponds to it. In such case we can find the probability of A by performing the integration over 5A, i.e., P(A) 33 f (x, y) dx dy
(21)
The joint distribution function of X and Y in this case is defined by
x y `
F(x, y)
x, Y
f (u, v) du dv
(22)
CHAPTER 2 Random Variables and Probability Distributions
It follows in analogy with (11), page 38, that '2F 'x 'y f (x, y)
(23)
i.e., the density function is obtained by differentiating the distribution function with respect to x and y. From (22) we obtain
F1(x)
3 ` v
f (u, v) du dv f (u, v) du dv
(24)
F2( y)
3 ` v
(25)
We call (24) and (25) the marginal distribution functions, or simply the distribution functions, of X and Y, respectively. The derivatives of (24) and (25) with respect to x and y are then called the marginal density functions, or simply the density functions, of X and Y and are given by
f1(x)
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