Lecture 5

You might compare the answer to Examples Sheet 1 #10 to a Bonferroni inequality working in the reverse direction, $\geq$.

We saw by examples today that it is possible to have three events $A_1,A_2,A_3$ such that every pair of events is independent, but the three events are not mutually independent. Here is another example. Suppose the sample sample consists of 4 equally likely outcomes $\{w_1,w_2,w_3,w_4\}$. Let

$A_1=\{w_1,w_2\}$,
$A_2=\{w_1,w_3\}$,
$A_3=\{w_1,w_4\}$.

Then $P(A_i\cap A_j)=P(A_i)P(A_j)=1/4$, but $1/4=P(A_1\cap A_2\cap A_3)\neq P(A_1)P(A_2)P(A_3)=1/8$.

We might generalize this to a puzzle. Can you find five events $A_1,A_2,A_3,A_4,A_5$ such that every subset of either two, three or four events is a set of mutually independent events, but the five events are not mutually independent?

Last year a student told me this nice answer. Roll a die 4 times. Let $A_i$ be the probability that the $i$th dice roll is odd, $i=1,2,3,4$ and let $A_5$ be the event the sum of the four rolls is even.

You can find another answer at the end of this page, but I think the above is nicer.

A famous example of data that is Poisson distributed is due to von Bortkiewicz (1898), who studied the numbers of Prussian cavalryman being killed by the kick of a horse in each of 20 years. This is an example with $n$ (number of cavalryman) large, and $p$ (chance a particular cavalryman being killed) small. You can read more about this famous example here.