Discrete distributions
The discrete uniform, Bernoulli, binomial, geometric and Poisson distributions: the situation each one models, its parameters, its formula, its expected value and variance, and the approximation of a binomial by a Poisson distribution.
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Discrete uniform distribution: what is it?
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The distribution of a variable that takes n possible values, all with the same probability 1/n: the result of a fair die.
It is the model of equal likelihood: for a six-sided die, each face has probability 1/6 and the expected value is 3.5.
A fair coin is tossed 5 times. What is the probability of exactly 3 heads?
0.31250.1250.031250.6
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0.3125C(5, 3) × 0.5³ × 0.5² = 10 / 32 = 0.3125. Forgetting the binomial coefficient gives 1/32 = 0.03125; 0.125 is the probability of three heads in a row on three tosses, and 0.6 is the proportion 3/5, which is not a probability.
When n is large and p small, the binomial distribution B(n; p) is close to the Poisson distribution with parameter ….
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When n is large and p small, the binomial distribution B(n; p) is close to the Poisson distribution with parameter np.
The Poisson parameter takes over the expected value of the binomial. With B(300; 0.01), approximate by a Poisson distribution with parameter 3.
Bernoulli distribution: what is it?
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The distribution of an experiment with two outcomes, success with probability p and failure with 1 − p, coded 1 and 0.
Its expected value is p and its variance p(1 − p). A network packet lost or not, a request failing or not, are examples.