Conditional Probability

$P(\cdot | B)$ is itself a valid probability measure on $(\Omega, \mathcal{F})$. In particular $P(B|B) = 1$ and $P($ satisfies all Kolmogorov axioms. The map is the original measure restricted and renormalized to .

The partition $(B_n)$ must be exhaustive (covers $\Omega$) and pairwise disjoint. In practice, choose a partition that makes each $P(A|B_n)$ easy to compute.

Whenever a problem involves multiple scenarios or causes, condition on them using the total probability formula. The key pattern is: identify a natural partition $(B_n)$, compute $P(A|B_n)$ in each case, then average weighted by $P($.

In Bayesian language: $P(B_k)$ is the prior, $P(A|B_k)$ is the likelihood, and $P$ is the . Bayes' formula inverts the conditioning — it answers "given that occurred, which cause is most likely?"

Bayes is the fundamental tool for inference: you observe data $A$ and update beliefs about a hidden cause $B_k$. Classic interview problems: medical tests (false positives), coin fairness, signal detection.

$E[X|Y]$ is a random variable — a function of $Y$, not a number. It becomes a number only once $Y$ is observed. This distinction is crucial: $E[X|Y=y]$ is a number, is a random variable.

The tower property is the single most used tool for computing expectations in multi-stage problems. Pattern: condition on the first source of randomness, compute the inner expectation, then take the outer expectation. Essential for: compound distributions, sequential experiments, Wald's identity.

The pull-out property $E[X \cdot g(Y)|Y] = g(Y) \cdot E[X|Y]$ is used constantly — it says that anything that is already known given $Y$ can be factored out of the conditional expectation.

The decomposition reads: total variance = average of conditional variances + variance of conditional means. The first term captures the within-group variability, the second captures the between-group variability.

Law of total variance is the key tool for compound/mixture distributions. Whenever $X$ depends on a hidden random variable $Y$ (number of events, regime, group), decompose using this law.

$\mathcal{G}$ represents the information available. The larger $\mathcal{G}$ is, the more information we have:

• $\mathcal{G} = \{\emptyset, \Omega\}$ (trivial): $E[X|\mathcal{G}] = E[X]$ — no information

The tower property in this form — $E[E[X|\mathcal{G}_2]|\mathcal{G}_1] = E[X|\mathcal{G}_1]$ for — is the exact property used to define martingales. A process is a martingale if for all .