Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
MISC. Simulating Normal Random Variables Simulation can provide a great deal of information about the behavior of a random variable Simulating Normal Random Variables • Two types of simulations (1) Generating fixed values - Uses Random Number Generation (2) Generating changeable values - Uses NORMINV function Simulating Normal Random Variables • Fixed Values • Random Number Generation is found under Data/Data Analysis • Values will never change • Useful if you need to show how your specific results are tabulated Simulating Normal Random Variables • Sample: Number of columns Number of rows Type of distribution Mean Standard Deviation (Leave blank) Cell where data is placed Simulating Normal Random Variables • Ex. Generate a fixed sample of data containing 25 values that has a normal distribution with a mean of 13 and a standard deviation of 4.6 Simulating Normal Random Variables • Soln: Integration Applications-oct1st • FundamentalTheorem Theorem ofof Calculus. For many of the functions, f, Fundamental Calculus x which occur in business applications, the derivative of - f (u) du, with a respect to x, is f(x). This holds for any number a and any x, such that the closed interval between a and x is in the domain of f. Example : applies to p.d.f.’s and c.d.f.’s Recall from Math 115a b a f X x dx FX b FX a 8 Integration, Calculus the inverse connection between integration and differentiation is called the Fundamental Theorem of Calculus. Fundamental Theorem of Calculus. For many of the functions, f, x which occur in business applications, the derivative of f (u) du, with a respect to x, is f(x). This holds for any number a and any x, such that the closed interval between a and x is in the domain of f. Example 7. Let f(u) = 2 for all values of u. If x 1, then integral of f from 1 to x is the area of the region over the interval [1, x], between the u-axis and the graph of f. 9 Integration, Calculus 3 (1, 2) 2 (x, 2) The region whose area is represented by the integral is rectangular, with height 2 and width x 1. Hence, its area is 2(x 1) = 2x 2, and x f (u ) 1 f (u) du 2 1 x1 0 x 0 1 2 3 u x 4 5 f (u) du 2 x 2. 1 In the section Properties and Applications of Differentiation, we saw that the derivative of f(x) = mx + b is equal to m, for all values of x. Thus, the x derivative of f (u) du, with respect to x, is equal to 2. As predicted by the 1 Fundamental Theorem of Calculus, this is also the value of f(x). The next example uses the definition of a derivative as the limit of difference quotients. 10