PX284 - A3 - the bernoulli trial

an independent experiment with two outcomes with the probabilities:

P_{i} (x) = {\begin{cases} p & x_{i} = 1 & ^{'} {success}^{'} \\ 1 - p & x_{i} = 0 & ^{'} {failure}^{'} \end{cases}

eg: tossing a coin
the mean is given by:

⟨ x ⟩ = \sum x_{i} P_{i} = 1 \times p + 0 \times (1 - p) = p

the mean of $x^{2} :$

⟨ x^{2} ⟩ = \sum x_{i}^{2} P_{i} = 1^{2} \times p + 0^{2} \times (1 - p) = p

the standard deviation:

σ_{x} = \sqrt{⟨ x^{2} ⟩ - ⟨ x ⟩^{2}} = \sqrt{p - p^{2}} = \sqrt{p (1 - p)}

if the same experiment is performed $n$ times, the probability of getting $k$ successes and $n - k$ failures is given by the binomial distribution:

P (n, k) =^{n} C_{k} p^{k} (1 - p)^{n - k}

where, $^{n} C_{k} = (\binom{n}{k}) = \frac{n!}{k! (n - k)!}$

the expected value and the standard deviation are:

\begin{matrix} ⟨ k ⟩ = n p \\ σ_{k} = \sqrt{n p (1 - p)} \end{matrix}

the fractional width is given by:

\frac{σ_{k}}{⟨ k ⟩} = \sqrt{\frac{1 - p}{n p}}

the fractional width $\to 0 as n \to \infty$

stirling's formula

it is an approximation that allows factorial calculations in the limit of large $n :$

\ln n! \approx n \ln n - n