Initial commit

Poisson Distribution.md

@@ -0,0 +1,97 @@
+#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.