97 lines
1.9 KiB
Markdown
97 lines
1.9 KiB
Markdown
#Math #Probability
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# The Poisson Distribution
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The Poisson Distribution describes a distribution where an event occurs for an interval of time, where there is an a mean number of times the event happens in the same interval of time.
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# Binomial Distribution to Poisson Distribution
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Binomial Distribution
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$$
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\frac {n!} {k!(n-k)!} p^k (1-p)^{n-k}
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$$
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Binomial Distribution with infinite trials
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$$
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\lim _{n\to\infty} \frac {n!} {k!(n-k)!} p^k (1-p)^{n-k}
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$$
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Let a be np, the average success rate in n intervals. This gives us the Poisson Distribution in another form.
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$$
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\lim _{n\to\infty} \frac {n!} {k!(n-k)!} (\frac {a} {n})^k (1-\frac {a} {n})^{n-k}
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$$
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$$
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\lim _{n\to\infty} \frac {n!} {k!(n-k)!} (\frac {a^k} {n^k}) (1-\frac {a} {n})^n(1-\frac {a} {n})^{-k}
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$$
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$$
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\frac {a^k} {k!} \lim _{n\to\infty} \frac {n!} {n^a(n-k)!} (1-\frac {a} {n})^n(1-\frac {a} {n})^{-k}
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$$
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Now we have three limits to evaluate
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# Evaluating the Limits
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## First Limit
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$$
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\lim _{n \to\infty} \frac {n!} {n^k(n-k)!}
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$$
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$$
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\lim _{n \to\infty} \frac {n(n-1)(n-2)...(n-k)(n-k-1)...(1)} {n^k(n-k)(n-k-1)...(1)}
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$$
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$$
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\lim _{n\to\infty} \frac {n(n-1)...(n-k+1)} {n^k}
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$$
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$$
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\lim _{n\to\infty} (\frac {n} {n})(\frac {n-1} {n})...(\frac {n-k+1} {n})
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$$
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As n goes to infinity, all the terms tend to 1. Therefore, the limit tends to 1.
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## Second Limit
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$$
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\lim _{n\to\infty} (1-\frac {a} {n})^n
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$$
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Let u be -n/x (note this tends to negative infinity)
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$$
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\lim _{n\to\infty}(1+\frac {1} {u})^{-au}
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$$
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Use definition of e
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$$
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e^{-a}
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$$
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## Third Limit
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$$
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\lim _{n\to\infty}(1-\frac{a} {n})^{-k}
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$$
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a/n tends to 0
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$$
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1^k
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$$
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Therefore this limit tends to 1.
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# Putting it together
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$$
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\frac {e^{-a}a^{k}}{k!}
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$$
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is the formula for the probability of an event happening k times in an interval of time, where a is the mean number of times of the event happening in the interval of time the event ran in. This is the formula for the Poisson Distribution. |