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
  • $P(T_1<T_2)=\frac{{\lambda }_1}{{\lambda }_1+{\lambda }_2}$
  • By integrating over $0\to \infty$ and $0\to t_2$.

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• Covariance measures linear association
• 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