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#Math #Probability
The Poisson Distribution
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.
Binomial Distribution to Poisson Distribution
Binomial Distribution
\frac {n!} {k!(n-k)!} p^k (1-p)^{n-k}
Binomial Distribution with infinite trials
\lim _{n\to\infty} \frac {n!} {k!(n-k)!} p^k (1-p)^{n-k}
Let a be np, the average success rate in n intervals. This gives us the Poisson Distribution in another form.
\lim _{n\to\infty} \frac {n!} {k!(n-k)!} (\frac {a} {n})^k (1-\frac {a} {n})^{n-k}
\lim _{n\to\infty} \frac {n!} {k!(n-k)!} (\frac {a^k} {n^k}) (1-\frac {a} {n})^n(1-\frac {a} {n})^{-k}
\frac {a^k} {k!} \lim _{n\to\infty} \frac {n!} {n^a(n-k)!} (1-\frac {a} {n})^n(1-\frac {a} {n})^{-k}
Now we have three limits to evaluate
Evaluating the Limits
First Limit
\lim _{n \to\infty} \frac {n!} {n^k(n-k)!}
\lim _{n \to\infty} \frac {n(n-1)(n-2)...(n-k)(n-k-1)...(1)} {n^k(n-k)(n-k-1)...(1)}
\lim _{n\to\infty} \frac {n(n-1)...(n-k+1)} {n^k}
\lim _{n\to\infty} (\frac {n} {n})(\frac {n-1} {n})...(\frac {n-k+1} {n})
As n goes to infinity, all the terms tend to 1. Therefore, the limit tends to 1.
Second Limit
\lim _{n\to\infty} (1-\frac {a} {n})^n
Let u be -n/x (note this tends to negative infinity)
\lim _{n\to\infty}(1+\frac {1} {u})^{-au}
Use definition of e
e^{-a}
Third Limit
\lim _{n\to\infty}(1-\frac{a} {n})^{-k}
a/n tends to 0
1^k
Therefore this limit tends to 1.
Putting it together
\frac {e^{-a}a^{k}}{k!}
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.