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Express the cdf of U := max(X,Y ) in the form P((X,Y ) Au ) for some set Au . in .NET Deploy UCC - 12 in .NET Express the cdf of U := max(X,Y ) in the form P((X,Y ) Au ) for some set Au .

Example 7.3. Express the cdf of U := max(X,Y ) in the form P((X,Y ) Au ) for some set Au . using barcode drawer for .net framework control to generate, create upca image in .net framework applications. UCC-14 7.1 Joint and marginal pro UPC A for .NET babilities Solution.

To nd the cdf of U, begin with FU (u) = P(U u) = P(max(X,Y ) u). Since the larger of X and Y is less than or equal to u if and only if X u and Y u, P(max(X,Y ) u) = P(X u,Y u) = P((X,Y ) Au ),. where Au := {(x, y) : x u and y u} is the shaded southwest region shown in Figure 7.3(a)..

u v u Figure 7.3. (a) Southwest region {(x, y) : x u and y u}.

(b) The region {(x, y) : x v or y v}.. Example 7.4. Express the c UPCA for .

NET df of V := min(X,Y ) in the form P((X,Y ) Av ) for some set Av . Solution. To nd the cdf of V , begin with FV (v) = P(V v) = P(min(X,Y ) v).

Since the smaller of X and Y is less than or equal to v if and only either X v or Y v, P(min(X,Y ) v) = P(X v or Y v) = P((X,Y ) Av ), where Av := {(x, y) : x v or y v} is the shaded region shown in Figure 7.3(b)..

Product sets and marginal probabilities The Cartesian product of two univariate sets B and C is de ned by B C := {(x, y) : x B and y C}. In other words, (x, y) B C x B and y C..

Bivariate random variables For example, if B = [1, 3] and C = [0.5, 3.5], then B C is the rectangle [1, 3] [0.

5, 3.5] = {(x, y) : 1 x 3 and 0.5 y 3.

5}, which is illustrated in Figure 7.4(a). In general, if B and C are intervals, then B C is a rectangle or square.

If one of the sets is an interval and the other is a singleton, then the product set degenerates to a line segment in the plane. A more complicated example is shown in Figure 7.4(b), which illustrates the product [1, 2] [3, 4] [1, 4].

Figure 7.4(b) also illustrates the general result that distributes over ; i.e.

, (B1 B2 ) C = (B1 C) (B2 C).. 4 3 2 1 0 4 3 2 1 0. 0 1 2 3 4 (a). 0 1 2 3 4 (b). Figure 7.4. The Cartesian products (a) [1, 3] [0.

5, 3.5] and (b) [1, 2] [3, 4] [1, 4]..

Using the notion of produc GTIN - 12 for .NET t set, {X B,Y C} = { : X( ) B and Y ( ) C} = { : (X( ),Y ( )) B C}, for which we use the shorthand {(X,Y ) B C}. We can therefore write P(X B,Y C) = P((X,Y ) B C).

The preceding expression allows us to obtain the marginal probability P(X B) as follows. First, for any event E, we have E , and therefore, E = E . Second, Y is assumed to be a real-valued random variable, i.

e., Y ( ) IR for all . Thus, {Y IR} = .

Now write P(X B) = P({X B} ) = P({X B} {Y IR}) = P(X B,Y IR) = P((X,Y ) B IR). Similarly, P(Y C) = P((X,Y ) IR C). (7.

1). 7.1 Joint and marginal pro babilities Joint cumulative distribution functions The joint cumulative distribution function of X and Y is de ned by FXY (x, y) := P(X x,Y y). We can also write this using a Cartesian product set as FXY (x, y) = P((X,Y ) ( , x] ( , y]).

. In other words, FXY (x, y) .NET GS1 - 12 is the probability that (X,Y ) lies in the southwest region shown in Figure 7.5(a).

. ( x, y ). (a) ( b). Figure 7.5. (a) Southwest region ( , x] ( , y].

(b) Rectangle (a, b] (c, d].. The joint cdf is important Visual Studio .NET GS1 - 12 because it can be used to compute P((X,Y ) A) for any set A. For example, you will show in Problems 3 and 4 that P(a < X b, c < Y d), which is the probability that (X,Y ) belongs to the rectangle (a, b] (c, d] shown in Figure 7.

5(b), is given by the rectangle formula2 FXY (b, d) FXY (a, d) FXY (b, c) + FXY (a, c). (7.2).

Example 7.5. If X and Y ha ve joint cdf FXY , nd the joint cdf of U := max(X,Y ) and V := min(X,Y ).

Solution. Begin with FUV (u, v) = P(U u,V v). From Example 7.

3, we know that U = max(X,Y ) u if and only if (X,Y ) lies in the southwest region shown in Figure 7.3(a). Similarly, from Example 7.

4, we know that V = min(X,Y ) v if and only if (X,Y ) lies in the region shown in Figure 7.3(b). Hence, U u and V v if and only if (X,Y ) lies in the intersection of these two regions.

The form of this intersection depends on whether u > v or u v. If u v, then the southwest region.
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