Raising the bar Law of total probability - you could have invented this

May 18

Law of total probability - you could have invented this

category: AP Stats and Business Stats, FN3142 Quantitative Finance, Quantitative Finance, Statistics 1, Statistics 2 No Comments

Law of total probability - you could have invented this

A knight wants to kill (event $K$ ) a dragon. There are two ways to do this: by fighting (event $F$ ) the dragon or by outwitting ( $O$ ) it. The choice of the way ( $F$ or $O$ ) is random, and in each case the outcome ( $K$ or not $K$ ) is also random. For the probability of killing there is a simple, intuitive formula:

$P(K)=P(K|F)P(F)+P(K|O)P(O)$ .

Its derivation is straightforward from the definition of conditional probability: since $F$ and $O$ cover the whole sample space and are disjoint, we have by additivity of probability

$P(K)=P(K\cap(F\cup O))=P(K\cap F)+P(K\cap O)=\frac{P(K\cap F)}{P(F)}P(F)+\frac{P(K\cap O)}{P(O)}P(O)$

$=P(K|F)P(F)+P(K|O)P(O)$ .

This is easy to generalize to the case of many conditioning events. Suppose $A_1,...,A_n$ are mutually exclusive (that is, disjoint) and collectively exhaustive (that is, cover the whole sample space). Then for any event $B$ one has

$P(B)=P(B|A_1)P(A_1)+...+P(B|A_n)p(A_n)$ .

This equation is call the law of total probability.

Application to a sum of continuous and discrete random variables

Let $X,Y$ be independent random variables. Suppose that $X$ is continuous, with a distribution function $F_X$ , and suppose $Y$ is discrete, with values $y_1,...,y_n$ . Then for the distribution function of the sum $F_{X+Y}$ we have