Three random variables independence

Asked Jun 23 '19 at 10:35

Active Jun 23 '19 at 10:35

Viewed 1,313 times

Is it possible to have such random variables $X, Y, Z$ (for simplicity let them be discrete) that: $$P(X \cap Y) = P(X)P(Y)$$ $$P(X \cap Z) = P(X)P(Z)$$ $$P(Y \cap Z) = P(Y)P(Z)$$

but

$$P(X \cap Y \cap Z) \neq P(X)P(Y)P(Z)$$

in other words $(X, Y)$, $(X, Z)$ and $(Y,Z)$ are independent, but $(X,Y,Z)$ as a group of three random variables is not ?

asked Jun 23 '19 at 10:35

Zarrie

3

The usual example is to flip a fair penny and a fair dime, let $X$ be the indicator variable for the penny coming up $H$, $Y$ the variable for the dime coming up $H$, and $Z$ the variable for the two coins matching. – lulu Jun 23 '19 at 10:38
Thanks, lulu! Pretty simple and clear example! – Zarrie Jun 23 '19 at 10:40
You use the word "variables" but your probabilities are written as if $X$, $Y$, and $Z$ are events. – David K Jun 23 '19 at 10:52
See also Example of Pairwise Independent but not Jointly Independent Random Variables? – Minus One-Twelfth Jun 23 '19 at 11:22

Three random variables independence

0 Answers0