ssrs 2008 r2 barcode font In (36) the Jacobian determinant, or briefly Jacobian, is given by 'x 'u 'y 'u 'x 'v 'y 'v in Software

Encoder QR Code JIS X 0510 in Software In (36) the Jacobian determinant, or briefly Jacobian, is given by 'x 'u 'y 'u 'x 'v 'y 'v

In (36) the Jacobian determinant, or briefly Jacobian, is given by 'x 'u 'y 'u 'x 'v 'y 'v
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'(x, y) '(u, v)
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(37)
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Probability Distributions of Functions of Random Variables
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Theorems 2-2 and 2-4 specifically involve joint probability functions of two random variables. In practice one often needs to find the probability distribution of some specified function of several random variables. Either of the following theorems is often useful for this purpose. X (the second choice is Theorem 2-5 Let X and Y be continuous random variables and let U 1(X, Y ), V arbitrary). Then the density function for U is the marginal density obtained from the joint density of U and V as found in Theorem 2-4. A similar result holds for probability functions of discrete variables. Theorem 2-6 Let f (x, y) be the joint density function of X and Y. Then the density function g(u) of the random variable U 1(X, Y ) is found by differentiating with respect to u the distribution
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CHAPTER 2 Random Variables and Probability Distributions
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function given by G(u) P[f1 (X, Y )
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(38)
Where 5 is the region for which
Convolutions
As a particular consequence of the above theorems, we can show (see Problem 2.23) that the density function of the sum of two continuous random variables X and Y, i.e., of U X Y, having joint density function f (x, y) is given by
g(u)
f (x, u
x) dx f1 (x)f2 (y), and (39) reduces to x) dx
(39)
In the special case where X and Y are independent, f(x, y)
g(u)
f1(x) f2 (u
(40)
which is called the convolution of f1 and f2, abbreviated, f1 * f2. The following are some important properties of the convolution: 1. f1 * f2 f2 * f1 2. f1 *( f2 * f3) ( f1 * f2) * f3 3. f1 *( f2 f3) f1 * f2 f1 * f3 These results show that f1, f2, f3 obey the commutative, associative, and distributive laws of algebra with respect to the operation of convolution.
Conditional Distributions
We already know that if P(A) 0, P(B u A) P(A B) P(A) x), (B: Y y), then (41) becomes (42) (41)
If X and Y are discrete random variables and we have the events (A: X P(Y where f (x, y) P(X for X. We define x, Y yuX x) f (x, y) f1(x)
y) is the joint probability function and f1 (x) is the marginal probability function f (y u x) ; f (x, y) f1(x)
(43)
and call it the conditional probability function of Y given X. Similarly, the conditional probability function of X given Y is f (x u y) ; f (x, y) f2(y) (44)
We shall sometimes denote f (x u y) and f( y u x) by f1 (x u y) and f2 ( y u x), respectively. These ideas are easily extended to the case where X, Y are continuous random variables. For example, the conditional density function of Y given X is f (y u x) ; f (x, y) f1(x) (45)
CHAPTER 2 Random Variables and Probability Distributions
where f (x, y) is the joint density function of X and Y, and f1 (x) is the marginal density function of X. Using (45) we can, for example, find that the probability of Y being between c and d given that x X x dx is P(c Y dux X x dx) 3c f ( y u x) dy
(46)
Generalizations of these results are also available.
Applications to Geometric Probability
Various problems in probability arise from geometric considerations or have geometric interpretations. For example, suppose that we have a target in the form of a plane region of area K and a portion of it with area K1, as in Fig. 2-5. Then it is reasonable to suppose that the probability of hitting the region of area K1 is proportional to K1. We thus define
Fig. 2-5
P(hitting region of area K1)
K1 K
(47)
where it is assumed that the probability of hitting the target is 1. Other assumptions can of course be made. For example, there could be less probability of hitting outer areas. The type of assumption used defines the probability distribution function.
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