Chapter5

# Chapter5 - Chapter 5 Important Distributions and Densities...

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Chapter 5 Important Distributions and Densities 5.1 Important Distributions In this chapter, we describe the discrete probability distributions and the continuous probability densities that occur most often in the analysis of experiments. We will also show how one simulates these distributions and densities on a computer. Discrete Uniform Distribution In Chapter 1, we saw that in many cases, we assume that all outcomes of an exper- iment are equally likely. If X is a random variable which represents the outcome of an experiment of this type, then we say that X is uniformly distributed. If the sample space S is of size n , where 0 <n< , then the distribution function m ( ω ) is deﬁned to be 1 /n for all ω S . As is the case with all of the discrete probabil- ity distributions discussed in this chapter, this experiment can be simulated on a computer using the program GeneralSimulation . However, in this case, a faster algorithm can be used instead. (This algorithm was described in Chapter 1; we repeat the description here for completeness.) The expression 1+ b n ( rnd ) c takes on as a value each integer between 1 and n with probability 1 /n (the notation b x c denotes the greatest integer not exceeding x ). Thus, if the possible outcomes of the experiment are labelled ω 1 ω 2 , ..., ω n , then we use the above expression to represent the subscript of the output of the experiment. If the sample space is a countably inﬁnite set, such as the set of positive integers, then it is not possible to have an experiment which is uniform on this set (see Exercise 3). If the sample space is an uncountable set, with positive, ﬁnite length, such as the interval [0 , 1], then we use continuous density functions (see Section 5.2). 183

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184 CHAPTER 5. DISTRIBUTIONS AND DENSITIES Binomial Distribution The binomial distribution with parameters n , p , and k was deﬁned in Chapter 3. It is the distribution of the random variable which counts the number of heads which occur when a coin is tossed n times, assuming that on any one toss, the probability that a head occurs is p . The distribution function is given by the formula b ( n,p,k )= ± n k p k q n - k , where q =1 - p . One straightforward way to simulate a binomial random variable X is to compute the sum of n independent 0 - 1 random variables, each of which take on the value 1 with probability p . This method requires n calls to a random number generator to obtain one value of the random variable. When n is relatively large (say at least 30), the Central Limit Theorem (see Chapter 9) implies that the binomial distribution is well-approximated by the corresponding normal density function (which is deﬁned in Section 5.2) with parameters μ = np and σ = npq . Thus, in this case we can compute a value Y of a normal random variable with these parameters, and if - 1 / 2 Y<n +1 / 2, we can use the value b Y / 2 c to represent the random variable X .I f Y< - 1 / 2or Y>n / 2, we reject Y and compute another value. We will see in the next section how we can quickly simulate normal random variables.
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## This note was uploaded on 06/23/2008 for the course STAT 601 taught by Professor Wherly during the Spring '08 term at Texas A&M.

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Chapter5 - Chapter 5 Important Distributions and Densities...

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