1. Math 4030 – 4a Discrete Distributions
Binomial
Hypergeometric
Poisson
Geometric

2. Hypergeometric Distribution (Sec. 4.3)
There are N units, of which a units are defective.
Randomly sample n units without replacement, and let X be the number of defective units in the sample.
Then
Sampling without replacement, and N is not large enough.
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3. Probability P(X=x) = h(x; n, a, N)?
Mean:
Variance:
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4. Binomial vs. Hypergeometric
N - a
n - x x a
In a pool of N objects, a are marked with an “S”.
Randomly select n.
X is the number of S’s in the sample of n.
X ~ H(n, a, N)
All possible values for X:
In an infinitely large pool, p100% are marked with an “S”.
Randomly select n.
X is the number of S’s in the sample of n.
X ~ Bi(n, p)
All possible values for X:
X = 0, 1, 2, …, n
0  x  n
0  x  a
0  n – x  N - a
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5. Binomial vs. Hypergeometric
X ~ Bi(n, p)
X ~ H(n, a, N)
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6. Poisson Distribution (Sec. 4.6)
Random variable X follows Poisson distribution, or X ~ Poisson(), if its probability distribution has the following formula
where  > 0 is the parameter (both mean and variance).
The cumulative distribution
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7. Poisson Processes (Sec. 4.7)
Consider independent and random “customer” arrivals over a given time interval. Let X be the number of arrivals. Then X has the Poisson distribution with parameter  as the mean/average number of arrivals on the interval.
phone calls at customer service;
students’ visit during office hour;
machine breakdown;
forest fire;
earthquake
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8. Geometric Distribution (Sec. 4.8)
A quality inspector inspects the electrical switches right off the manufacturing belt. He is interested in the question: How may items are to be inspected until the first failure occurs?
P(X=x) = g(x;p) = p(1-p)x-1, for x = 1,2,….
Cumulative probability:
The mean and the variance:

9. Negative Binomial Distribution NB(r,p):
A quality inspector inspects the electrical switches right off the manufacturing belt. He is interested in the question: How may items are to be inspected until the r failures are found?
When r = 1, we have geometric distribution.

10. Binomial? Hypergeometric? Poisson? Or Geometric?
Binomial – Sample with replacement: n trials, each with two outcomes (S or F), identical probability p, independent. X is the number of “S” in n trials.
Hypergeometric – Sample without replacement: N objects, of them a have marked with “S”. Take a sample of size n (without replacement). X is the number of “S” in the sample.
Poisson – Number of arrivals: In a given time period, there are  independent arrivals on average. X is the actual number of arrivals in any given time period.
Geometric – When to get the first “S”: Repeat the independent Bernoulli trials until the first “S” occurs. X is the number of trials repeated.
Negative Binomial – When to get the r-th “S”: Repeat the independent Bernoulli trials until the exactly r “S” occurs. X is the number of trials repeated.

11. Chebyshev’s Theorem (Sec.4.5)
If X is a random variable with mean  and standard deviation , then for any number k > 1,
Probability
Define the boundaries based on SD
Central location the distribution
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12. Application?
The number of customers who visit a car dealer’s showroom is a random variable with mean 18 and standard deviation 2.5. With what probability can we assert that there will be more than 8 but fewer than 28 customers?
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