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Statistical Inference: A Short Course by Michael J. Panik

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5.3 The Binomial Probability Distribution

While there are many discrete probability distributions, by far the most important one is the binomial distribution. To set the stage for a discussion of this distribution, we will need to cover two preliminary notions: (1) counting issues; and (2) the Bernoulli probability distribution.

Table 5.9 Weekly Lottery Prizes.

Number of Prizes Dollar Amount
    1 25,000
    4  5,000
  50    500
945       0

Table 5.10 Prize Probabilities.

X f(X)
25,000 0.001
  5,000 0.004
    500 0.050
        0 0.945
1.000

5.3.1 Counting Issues

We may generally view the concept of a permutation as an ordered set of objects. If any two of the objects are interchanged, we have a new permutation. For instance, the letters “a, b, c” form a particular permutation of the first three letters of the alphabet. If we interchange b and c, then “a, c, b” gives us another distinct permutation of these letters. More specifically, a permutation is any particular arrangement of r objects selected from a set of n distinct objects, rn.

What is the total number of permutations of r objects selected from a set of n distinct objects? To answer this, let us find

(5.7) equation

the number of permutations of n different objects taken r at a time, where

img

For instance, ...

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