1. MISC.

2. Simulating Normal Random Variables Simulation can provide a great deal of information about the behavior of a random variable

3. Simulating Normal Random Variables Two types of simulations (1) Generating fixed values - Uses Random Number Generation (2) Generating changeable values - Uses NORMINV function

4. 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

5. Simulating Normal Random Variables Sample: Number of columns Number of rows Type of distribution Mean Standard Deviation (Leave blank) Cell where data is placed

6. 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

7. Simulating Normal Random Variables Soln:

8. 8 Integration Applications-oct1st Fundamental Theorem of Calculus - Example : applies to p.d.f.’s and c.d.f.’s Recall from Math 115a Fundamental Theorem of Calculus. For many of the functions, f, which occur in business applications, the derivative of with 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.

9. 9 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, which occur in business applications, the derivative of with 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.

10. 10 Integration, Calculus 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) = 2  x  2, and (1, 2)(x, 2) x 2 x  1 In the section Properties and Applications of Differentiation, we saw that the derivative of f(x) = m  x + b is equal to m, for all values of x. Thus, the derivative of with respect to x, is equal to 2. As predicted by the 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.