2
May 18

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

$F_{X+Y}(t)=P(X+Y\le t)=\sum_{j=1}^nP(X+Y\le t|Y=y_j)P(Y=y_j)$

(by independence conditioning on $Y=y_j$ can be omitted)

$=\sum_{j=1}^nP(X\le t-y_j)P(Y=y_j)=\sum_{j=1}^nF_X(t-y_j)P(Y=y_j)$.

Compare this to the much more complex derivation in case of two continuous variables.

8
Oct 17

Reevaluating probabilities based on piece of evidence

This actually has to do with the Bayes' theorem. However, in simple problems one can use a dead simple approach: just find probabilities of all elementary events. This post builds upon the post on Significance level and power of test, including the notation. Be sure to review that post.

Here is an example from the guide for Quantitative Finance by A. Patton (University of London course code FN3142).

Activity 7.2 Consider a test that has a Type I error rate of 5%, and power of 50%.

Suppose that, before running the test, the researcher thinks that both the null and the alternative are equally likely.

1. If the test indicates a rejection of the null hypothesis, what is the probability that the null is false?

2. If the test indicates a failure to reject the null hypothesis, what is the probability that the null is true?

Denote events R = {Reject null}, A = {fAil to reject null}; T = {null is True}; F = {null is False}. Then we are given:

(1) $P(F)=0.5;\ P(T)=0.5;$

(2) $P(R|T)=\frac{P(R\cap T)}{P(T)}=0.05;\ P(R|F)=\frac{P(R\cap F)}{P(F)}=0.5;$

(1) and (2) show that we can find $P(R\cap T)$ and $P(R\cap F)$ and therefore also $P(A\cap T)$ and $P(A\cap F).$ Once we know probabilities of elementary events, we can find everything about everything.

Figure 1. Elementary events

Answering the first question: just plug probabilities in

$P(F|R)=\frac{P(R\cap F)}{P(R)}=\frac{P(R\cap F)}{P(R\cap T)+P(A\cap T)}.$

Answering the second question: just plug probabilities in

$P(T|A)=\frac{P(A\cap T)}{P(A)}=\frac{P(A\cap T)}{P(A\cap T)+P(A\cap F)}.$

Patton uses the Bayes' theorem and the law of total probability. The solution suggested above uses only additivity of probability.