f (x) dx + in .NET Paint Universal Product Code version A in .NET f (x) dx +

f (x) dx + generate, create upc code none on .net projects PDF-417 2d barcode 1 . f (x) dx,. where f (x) = ( / GS1 - 12 for .NET )/( 2 + x2 ) is the Cauchy density. Since the Cauchy density is an even function, / dx.

P(. X. 1) = 2 2 2 1 +x upc a for .NET Now make the change of variable y = x/ , dy = dx/ , to get P(. X. 1) = 2. 1/ . 1/ dy. 1 + y2 Since the integrand is nonnegative, the integral is maximized by minimizing 1/ or by maximizing . Hence, choosing = 10 maximizes your probability of winning..

4.1 Densities and probabilities we write f N(m, 2 ) if Gaussian / normal. T UPC Code for .NET he most important density is the Gaussian or normal.

For 2 > 0,. 1 1 x m f (x) = exp 2 2 (4.2). where is the posit ive square root of 2 . A graph of the N(m, 2 ) density is sketched in Figure 4.3.

It is shown in Problems 9 and 10 that the density is concave for x [m , m + ] and convex for x outside this interval. As increases, the height of the density decreases and it becomes wider as illustrated in Figure 4.5.

If m = 0 and 2 = 1, we say that f is a standard normal density.. = small = large 0 m Figure 4.5. N(m, 2 ) densities with different values of . As a consequence of the central limit theorem, whose discussion is taken up in 5, the Gaussian density is a good approximation for computing probabilities involving a sum of many independent random variables; this is true whether the random variables are continuous or discrete! For example, let X := X1 + + Xn , where the Xi are i.i.d.

with common mean m and common variance 2 . For large n, it is shown in 5 that if the Xi are continuous random variables, then fX (x) while if the Xi are integer-valued, 1 k nm 1 pX (k) exp 2 n 2 n. 1 x nm 1 exp 2 n 2 n In particular, since the macroscopic noise current measured in a circuit results from the sum of forces of many independent collisions on an atomic scale, noise current is well-described by the Gaussian density. For this reason, Gaussian random variables are the noise model of choice in electronic communication and control systems. To verify that an arbitrary normal density integrates to one, we proceed as follows.

(For an alternative derivation, see Problem 17.) First, making the change of variable t = (x m)/ shows that 2 1 e t /2 dt. f (x) dx = 2 .

Continuous random variables So, without loss of generality, we may assume f is a standard normal density with m = 0 2 and = 1. We then need to show that I := e x /2 dx = 2 . The trick is to show instead that I 2 = 2 .

First write I2 =. 2 /2. 2 /2. dy . Now write the product of integrals as the iterated integral I2 = e (x 2 +y2 )/2. dx dy. Next, we interpret t his as a double integral over the whole plane and change from Cartesian coordinates x and y to polar coordinates r and . To integrate over the whole plane in polar coordinates, the radius r ranges from 0 to , and the angle ranges from 0 to 2 . The substitution is x = r cos and y = r sin .

We also change dx dy to r dr d . This yields I2 = = =. 2 0 2 0 2 0 0 . 2 /2. r dr d e r 1 d 2 /2. = 2 . Example 4.6.

The noise voltage in a certain ampli er has the standard normal density. Show that the noise is as likely to be positive as it is to be negative. Solution.

In terms of the density, which we denote by f , we must show that. 0 0. f (x) dx = f (x) dx. 2 Since f (x) = e x /2 / 2 is an even function of x, the two integrals are equal. Furthermore, we point out that since the sum of the two integrals is f (x) dx = 1, each individual integral must be 1/2. Location and scale parameters and the gamma densities Since a probability density function can be any nonnegative function that integrates to one, it is easy to create a whole family of density functions starting with just one density function.

Let f be any nonnegative function that integrates to one. For any real number c and any positive number , consider the nonnegative function. f (x c) .. Here c is called a l ocation parameter and is called a scale parameter. To show that this new function is a probability density, all we have to do is show that it integrates to one. In the integral.

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