Some things to note: The sum of all the coefficients is \(2^{n}\). To prove this, evaluate \(\left(a+b\right)^{n}\) with \(a=1,b=1\) The...
\(z^{n}+\overline{z}^{n}\) is a real number for all positive \(n\). Proof 1 Treat \(z=re^{i\theta}\) and you can show that the sum...
Let \(X\) be the number of successes. If \(n<
\(\newcommand{\Cov}{\mathrm{Cov}}\) \(\newcommand{\Corr}{\mathrm{Corr}}\) \(\newcommand{\Sample}{X_{1},\dots,X_{n}}\) For any...
\(\newcommand{\Cov}{\mathrm{Cov}}\) \(\newcommand{\Corr}{\mathrm{Corr}}\) \(\newcommand{\Sample}{X_{1},\dots,X_{n}}\) The Method...
\(X\) is of a Poisson distribution if its pmf is \(p(x;\lambda)=\frac{e^{-\lambda}\lambda^{x}}{x!}\) where \(x\) is 0, 1, 2, etc....
The experiment requires: The trials be independent The outcome is binary (success or failure). The probability of success or failure is...
The hypergeometric experiment requires: The population is finite, with \(N\) individuals. The outcome of each trial is binary (success...
Bernoulli Distribution A Bernoulli random variable is one whose only possible values are 0 and 1. \(P(X=1)=p\) \(E[X]=p,V[X]=p(1-p)\)...