Category: probability and statistics

Interarrival times for Poisson distribution: Exponential distribution \begin{eqnarray} f(x) = \lambda e^{-\lambda x} \end{eqnarray} Its moment is \begin{eqnarray} M(s) = \frac{1}{1-\frac{s}{\lambda}} \end{eqnarray} Higher order Interarrival…

Exponential Family of distributions The probability distributions we have discussed in the previous chapter (and many other frequently used ones) are members of the exponential…

Bernoulli Trials A Bernoulli trial is basically a single coin toss or a dice roll. It generates an experimental value for a discrete random variable…

wikipedia example If we have a pdf $f_X(x)$, its moment generating function is defined as \begin{eqnarray} M_X(t)=E[e^{tX}] = \int_{-\infty}^{\infty} e^{tx} f_X(x) dx \end{eqnarray} On the…

Let $X$ be a random variable with pdf $f_X(x)$, and let $X$ be defined in the domain $x_0 \leq X \leq x_1$. Let $Y=g(X)$. Naturally,…

Sample space, samples, events, random experiments and probabilities. In this chapter we will give some basic definitions. Outcomes, sample space and Random experiments The setting…

Definition of exponentiation \begin{eqnarray} \lim_{n \rightarrow \infty}\left( 1+ \frac{1}{n} \right)^n =e\\ \lim_{n \rightarrow \infty}\left( 1+ \frac{x}{n} \right)^n =e^x \end{eqnarray} The second equation can easily be…