Types of RV’s and Properties

Tags	Properties

Bernoulli

Heads or tails, with success rate $p$ . (this is a piecewise function)

E[X] = p\\ Var[X] = p(1-p)

We can express the likelihood as a function (which allows for certain numerical analysis)

x \sim Ber(\phi) \rightarrow P(x) = \phi^x(1-\phi)^{(1-x)}

Binomial

This is the number of success in $n$ trials given success probability $p$ .

P(X = k) = \binom{n}{k}p^k(1-p)^{n-k}

Here are some facts

binomial RV is the sum of bernoulli RV's ( $Ber(p) = Bin(1, p)$ )

It follows that

E[X] = np\\ Var(X) = np(1-p)

Binomial distributions need not be symmetrical.

Poisson

This is the distribution of $x$ events happening given that $\lambda$ events usually happen. It is the limit of $Bin(n, p)$ such that $np = \lambda$ and $n→ \infty$ .

p(X = x) = \frac{\lambda^xe^{-\lambda}}{x!}

The stats are

E[X] = Var[X] = \lambda

Geometric

This is the number of trials until the first success

P(X = x) = (1-p)^{k-1}p

As such:

E[X] = \frac{1}{p}\\ Var(X) = \frac{1-p}{p^2}

Negative Binomial

This is the number of trials until $r$ successes.

P(X = k) = \binom{k-1}{r-1}(1-p)^{k-r}p^r

It is also the sum of $r$ geometric distributions, which means that

E[X] = \frac{r}{p}\\ Var(X) = \frac{r(1-p)}{p^2}

Uniform

This is just a distribution with uniform probability between $\alpha, \beta$ . The following are true:

E[X] = (\alpha + \beta) / 2 \\ Var(X) = (\beta - \alpha)^2 / 12

Exponential

This is the time until first success

f(X = x) = \lambda e^{-\lambda x}

The following are true:

E[X] = \frac{1}{\lambda} \\ Var(X) = \frac{1}{\lambda^2}

Poisson and Exponentials are deeply connected. The poisson is events in a time, and exponential is time until an event.