Bernoulli

\[ I \sim \text{Bernoulli}(p) \]

The Bernoulli probability distribution is

\[ P(I=k) = kp + (1-k)(1-p) \]

more naturally,

\(I=k\)	\(P(I=k)\)
\(1\)	\(p\)
\(0\)	\(1-p\)

Expected Value : $\( \mathbb E[I] = p \)$

Variance : $\( \text{Var}[I] = pq \)$

Covariance : $\( \text{Cov}[I_1,I_2] = P(I_1=1, I_2=1) -P(I_1=1)P(I_2=1) \)$

MGF : $\( M_X(t) = pe^t + q \)$