# Joint Distribution - Summing over the possible values of $Y$ = _marginalizing out_ $Y$ - Independence - conditional PDF = marginal PDF - joint PDF factors into marginal PDFs - The example of Chicken-Egg - $N \sim \text{Pois}(\lambda)$ eggs are laid, and hatched with probability of $p$ - The number of hatched eggs is $\text{Pois}(\lambda p)$, and that of the unhatched is $\text{Pois}(\lambda q)$. - The total number of eggs is random, hence the two r.v.s are counterintuitively independent. - The example of comparing exponentials - $P(T_1 < T_2) = \dfrac{\lambda_1}{\lambda_1 + \lambda_2}$ - By integrating over $0 \to \infty$ and $0 \to t_2$. --- - _Covariance_ measures _linear_ association - [[multinomial|Multinomial]] distribution - Multivariate Normal Distribution - All linear combination of the r.v.s in the vector must be normal. - In MVN, independence and zero correlation are equivalent