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Page 14-8 Chapter 15 Probability Distributions In this Chapter we provide examples of applications of the pre-defined probability distributions in the calculator. The MTH/PROBABILITY.. sub-menu -part 1 The MTH/PROBABILITY.. sub-menu is accessible through the keystroke sequence „.. With system flag 117 set to CHOOSE boxes, the following functions are available in the PROBABILITY.. menu: In this section we discuss functions COMB, PERM, ! (factorial), and RAND. Factorials, combinations, and permutations The factorial of an integer n is defined as: n! = n. (n-1) . (n-2)…3.2.1. By definition, 0! = 1. Factorials are used in the calculation of the number of permutations and combinations of objects. For example, the number of permutations of r objects from a set of n distinct objects is P = n(n . 1)(n . 1)...(n . r + 1) = n!/(n . r) nr Also, the number of combinations of n objects taken r at a time is . . .. . n . . .. . n(n .1)(n . 2)...(n . r + 1) n! = = r! r!(n . r)! r Page 15-1 We can calculate combinations, permutations, and factorials with functions COMB, PERM, and ! from the MTH/PROBABILITY.. sub-menu. The operation of those functions is described next: • COMB(n,r): Calculates the number of combinations of n items taken r at a time • PERM(n,r): Calculates the number of permutations of n items taken r at a time • n!: Factorial of a positive integer. For a non-integer, x! returns .(x+1), where .(x) is the Gamma function (see Chapter 3). The factorial symbol (!) can be entered also as the keystroke combination ~‚2. Example of applications of these functions are shown next: [Note: not all lines will be visible when done with the exercises in the following figure.] Random numbers The calculator provides a random number generator that returns a uniformly distributed random real number between 0 and 1. To generate a random number, use function RAND from the MTH/PROBABILITY sub-menu. The following screen shows a number of random numbers produced using RAND. (Note: The random numbers in your calculator will differ from these). Additional details on random numbers in the calculator are provided in Chapter 17 of the user's guide. Specifically, the use of function RDZ, to re- Page 15-2 start lists of random numbers is presented in detail in Chapter 17 of the user's guide. The MTH/PROB menu - part 2 In this section we discuss four continuous probability distributions that are commonly used for problems related to statistical inference: the normal distribution, the Student’s t distribution, the Chi-square (.2) distribution, and the F-distribution. The functions provided by the calculator to evaluate probabilities for these distributions are NDIST, UTPN, UTPT, UTPC, and UTPF. These functions are contained in the MTH/PROBABILITY menu introduced earlier in this chapter. To see these functions activate the MTH menu: „. and select the PROBABILITY option: The Normal distribution Functions NDIST and UTPN relate to the Normal distribution with mean . , and variance .2. To calculate the value of probability density function, or pdf, of the f(x) for the normal distribution, use function NDIST(.,.2,x). For example, check that for a normal distribution, NDIST(1.0,0.5,2.0) = 0.20755374. This function is useful to plot the Normal distribution pdf). The calculator also provides function UTPN that calculates the upper-tail normal distribution, i.e., UTPN(.,.2, x) = P(X>x) = 1 -P(X