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Discrete Uniform Distribution The discrete uniform distribution occurs when there are a finite number (m) of equally likely outcomes possible. The pmf of a uniform discrete random variable X is: p(x) = 1 / m, where x=1,2,…,m The mean and variance of a discrete uniform random variable X are: σ 2 = (m 2 - 1) / 12 µ = (m + 1) / 2

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Bernoulli Distribution A random experiment with two possible outcomes that are mutually exclusive and exhaustive is called a Bernoulli Trial. One outcome is arbitrarily labeled a “success” and the other a “failure” p is the probability of a success q = 1 - p is the probability of a failure The Bernoulli random variable X assigns: X(failure) = 0 and X(success) = 1

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Bernoulli Distribution The pmf for a Bernoulli random variable X is: p(x) = p x (1-p) 1-x, where x=0,1 The mean and variance of a Bernoulli random variable X are: µ = p σ 2 = pq

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Binomial Distribution A binomial experiment results from a sequence of n independent Bernoulli trials. The probability of success (p) remains constant in a binomial experiment The number of successes (X) is the random variable of interest in a binomial experiment If Y 1, Y 2, …, Y n are independent Bernoulli random variables, then X=∑ Y i is a binomial random variable.

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Binomial Distribution The pmf for a Binomial random variable X is: p(x) = n C x p x (1-p) n-x, where x=0,1,…,n The mean and variance of a binomial random variable X are: µ = np σ 2 = npq

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Hypergeometric Distribution The hypergeometric distribution applies when sampling without replacement from two possible mutually exclusive and exhaustive outcomes. Let X be the number of objects of type 1 drawn if n objects are drawn from N where there are M objects of type 1 and N-M objects of type 2. Then X is a hypergeometric random variable and the pmf of X is:

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Hypergeometric Distribution The mean and variance of a hypergeometric random variable X are: µ = n · (M/N) σ 2 = n · (M/N) · (1-M/N) · (N-n)/(N-1) As M and N converge to infinity and (M/N) converges to p, the hypergeometric distribution with n samples converges to the binomial distribution with n trials and p=M/N.

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Geometric Distribution A geometric distribution occurs when sampling independent Bernoulli trials. If X is the number of Bernoulli trials until the first success is observed, then X is a geometric random variable with pmf: p(x) = (1-p) x-1 p,where x=1,2,3,… The mean and variance of a geometric random variable X are: µ = 1 / p σ 2 = q / p 2

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Geometric Distribution Notice that for integer k, P( X > k ) = q k P( X ≤ k ) = 1 – q k “Memoryless” or “No Memory” Property If X is a geometric random variable, then P( X > j + k | X > j ) = P( X > k ) This implies that in independent Bernoulli trails, there is no such thing as being “due” to observe a success.

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Negative Binomial Distribution A negative binomial distribution occurs when sampling independent Bernoulli trials. If X is the number of Bernoulli trials until the r th success is observed, then X is a negative binomial random variable with pmf: p(x) = x-1 C r-1 p r (1-p) x-r,where x=r,r+1,r+2,… The mean and variance of a negative binomial random variable X are: µ = r (1/p) σ 2 = r (q / p 2 )

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Poisson Distribution The Poisson distribution describes the number of occurrences of an event in a given time or on a given interval. Assumptions of a Poisson Process 1. The number of events occurring in non- overlapping intervals is independent. 2. The probability of 1 event occurring in a significantly short interval h is h. 3. The probability of 2 events occurring in a significantly short interval h is essentially zero.

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Poisson Distribution If X is defined to be the number of occurrences of an event in a given continuous interval and is associated with a Poisson process with parameter >0, then X has a Poisson distribution with pdf: The mean and variance of a Poisson random variable X are: µ = 2 = µ = 2 =

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Poisson Distribution If events of a Poisson process occur at a mean rate of per unit, then the expected number of occurrences in an interval of length t is t. Moreover, if Y is the number of occurrences in an interval of length t, it is Poisson with pdf:

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Poisson Distribution The Poisson distribution with parameter =np is useful for approximating the binomial distribution with sample size n and probability of success p in cases with sufficiently large sample size (n>20 and p 20 and p<0.05). B(n,p) P( =np) as n ∞, p 0, and np and np

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