Distributions

Bernoulli Distribution

Def.: If $X$ has range $W_{X} = {0, 1}$ and density

f_{X} (α) = ⎩ ⎨ ⎧ p 1 - p 0 for α = 1 for α = 0 otherwise

$⟺$ $X \sim Bernoulli (p)$

Theorem

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

Def.: the distribution that follows from performing a Bernoulli event $X$ times.

f_{X} (k) = ⎩ ⎨ ⎧ (\frac{n}{k}) p^{k} (1 - p)^{n - k} 0 for k \in {0, 1, \dots, n} otherwise

$⟺$ $X \sim Bin (n, p)$

Eg.: Flip a coin $n$ times. $X :=$ # of heads

Pr [X = k] = (\frac{n}{k}) p^{k} (1 - p)^{n - k}

Intuition:

We have a pool of $n$ balls with indices on them. If we pick a ball, then the event we are counting occurred at that time.
We can pick any $k$ balls to get $X = k$

Limit of Binomial distribution

P (X = k) = \frac{λ ^{k} e ^{- λ}}{k !}

$k$ is the number of events you are counting. For example: number of emails you receive in an hour, or number of cars passing a junction in 10 minutes.
$λ$ (lambda) is the average rate or expected number of events in the given interval. For example: if you usually get 5 emails per hour on average, then $λ = 5$ .

$Pr [X = x] :=$ Probability that a certain Bernoulli event occurred after exactly $x$ tries, not before.

The probability of observing $k$ failures before achieving $r$ successes, where each trial has success probability $p$ :

P (X = k) = (k k + r - 1) p^{r} (1 - p)^{k}, k = 0, 1, 2, \dots

where:

Mean: $μ = \frac{r ( 1 - p )}{p}$

Variance: $σ^{2} = \frac{r ( 1 - p )}{p ^{2}}$

Total number successes of $n$ independent Bernoulli events.