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Chapter 3 Some Special Distributions Math 6203 Fall 2009 Instructor: Ayona Chatterjee

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3.1 The Binomial and Related Distributions

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Bernoulli Distribution A Bernoulli experiment is a random experiment in which the outcome can be classified in one of two mutually exclusive and exhaustive ways. – Example: defective/non-defective, success/failure. A sequence of independent Bernoulli trials has a fixed probability of success p.

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Let X be a random variable associated with a Bernoulli trial. X = 1 implies a success. X = 0 implies a failure. The pmf of X can be written as : p(x)=p x (1-p) 1-x x=0,1 In a sequence of n Bernoulli trials, we are often interested in a total of X number of successes.

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Binomial Experiment Independent and identical n trials. Probability of success p is fixed for each trial. Only two possible outcomes for each trial. Number of trails are fixed.

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Binomial Distribution The random variable X which counts the number of success of a Binomial experiment is said to have a Binomial distribution with parameters n and p and its pmf is given by:

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Theorem

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Negative Binomial Distribution Consider a sequence of independent repetitions of a random experiment with constant probability p of success. Let the random variable Y denote the total number of failures in this sequence before the rth success that is, Y +r is equal to the number of trials required to produce exactly r successes. The pmf of Y is called a Negative Binomial distribution

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Thus the probability of getting r-1 successes in the first y+r-1 trials and getting the rth success in the (y+r)th trial gives the pmf of Y as

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Geometric Distribution The special case of r = 1 in the negative binomial, that is finding the first success in y trials gives the geometric distribution. Thus we can re-write P(Y)=p q y-1 for y = 1, 2, 3, …. Lets find mean and variance for the Geometric Distribution.

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Multinomial Distribution Define the random variable X i to be equal to the number of outcomes that are elements of C i, i = 1, 2, … k-1. Here C1, C2, … Ck are k mutually exhaustive and exclusive outcomes of the experiment. The experiment is repeated n number of times. The multinomial distribution is

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Trinomial Distribution Let n= 3 in the multinomial distribution and we let X1 = X and X2= Y, then n –X-Y = X3 we have a trinomial distribution with the joint pmf of X and Y given as

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3.2 The Poisson Distribution A random variable that has a pmf of the form p(x) as given below is said to have a Poisson distribution with parameter m.

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Poisson Postulates Let g(x,w) denote the probability of x changes in each interval of length w. Let the symbol o(h) represent any function such that The postulates are – g(1,h)=λh+o(h), where λ is a positive constant and h > 0. – and – The number of changes in nonoverlapping intervals are independent.

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Note The number of changes in X in an interval of length w has a Poisson distribution with parameter m = wλ

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Theorem

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3.3 The Gamma, Chi and Beta Distributions The gamma function of α can be written as

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The Gamma Distribution A random variable X that has a pdf of the form below is said to have a gamma distribution with parameters α and β.

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Exponential Distribution The gamma distribution is used to model wait times. W has a gamma distribution with α = k and β= 1/ λ. If W is the waiting time until the first change, that is k = 1, the pdf of W is the exponential distribution with parameter λ and its density is given as

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Chi-Square Distribution A special case of the Gamma distribution with α=r/2 and β=2 gives the Chi-Square distribution. Here r is a positive integer called the degrees of freedom.

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Theorem

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Beta Distribution A random variable X is said to have a beta distribution with parameters α and β if its density is given as follows

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The Normal Distribution We say a random variable X has a normal distribution if its pdf is given as below. The parameters μ and σ 2 are the mean and variance of X respectively. We write X has N(μ,σ 2 ).

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The mgf The moment generating function for X~N(μ,σ 2 ) is

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Theorems

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3.6 t and F-Distributions

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The t-distribution Let W be a random variable with N(0,1) and let V denote a random variable with Chi- square distribution with r degrees of freedom. Then Has a t-distribution with pdf

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The F-distribution Consider two independent chi-square variables each with degrees of freedom r 1 and r 2. Let F = (U/r 1 )/(V/r 2 ) The variable F has a F-distribution with parameters r 1 and r 2 and its pdf is

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Student’s Theorem

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