Distribution

Bernoulli Distribution

The Bernoulli distribution is the discrete probability distribution of a Random Variable which takes the value $1$ with probability $p$ and the value $0$ with probability $q=1-p$.

• Exactly like Epsilon-Greedy algorithm

Bernoulli Distribution (Definition)

$X\sim Ber(p)$ if $X$ takes values $\{0,1\}$ with $P(X=1)=p$

The Bernoulli Distribution is a special case of the Binomial Distribution where a single trial is conducted (so $n=1$).

• Probability Mass Function
  • $f(0)=1-p$
  • $f(1)=p$

Expectation and Variance

• $E(X)=p$
• $Var(X)=p(1-p)$

Bernoulli Sequence

Suppose we have a sequence of independent $X_i\sim Ber(p)$, then we call $X_1,X_2,\dots ,X_n$ a Bernoulli process, (essentially just a sequence of 0s and 1s).