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  • STA256 Lecture 05

Statistical Independence

• Events $A$ and $B$ in sample space $S$ are independent if:
  • $P(A)>0$ and $P(B)>0$.
    • Or we would get division by zero
  • $P(A|B)=P(A)$
    • If they're independent, $B$ has no effect
    • $P(A|B)=P(A)$
    • $\frac{P(A\cap B)}{P(B)}=P(A)$
    • $P(A\cap B)=P(B)P(A)$
  • $P(B|A)=P(B)$
    • True for independent events
    • $\frac{P(B\cap A)}{P(A)}=P(B)$
    • $P(B\cap A)=P(A)P(B)$
  • With the above, we see that $P(A\cap B)=P(A)P(B)$.
  • To verify events are independent. Verify any of these identities.
• Example from Quiz 1: Question 5:
  • $PD$, Event Peter drinks coffee
  • $SD$, Event Sarah drinks coffee
  • Determine whether these are independent in the following:
    1. $P(PD)=0.4$, $P(SD)=0.5$, $P(PD\cup SD)=0.8$
       • By additive property:
         • $P(A\cup B)=P(A)+P(B)-P(A\cap B)$
         • $P(PD\cup SD)=P(PD)+P(SD)-P(PD\cap SD)$
         • $0.8=0.4+0.5-P(PD\cap SD)$
         • $0.8=0.9-P(PD\cap SD)$
         • $-0.1=-P(PD\cap SD)$
         • $0.1=P(PD\cap SD)$
       • Checking Independence:
         • $P(PD\cap SD)=P(PD)\cdot P(SD)$
         • $0.1=P(PD)\cdot P(SD)$
         • $0.1=0.4\cdot 0.5$
         • $0.1=0.2$
         • $0.1\ne 0.2$
         • So this is not independent
         • These are $\boxed{{\displaystyle \text{Dependent}}}$
         • So one event influence the other event
    2. $P(SD)=0.4$, $P(SD\cup PD)=0.28$, Probability of none is $0.18$
       • To get $P(PD)$, the $P(S)=1$
       • so $x+0.28+0.12+0.4+0.18$
       • $x+0.28+0.12+0.18=1$
       • $0.42$
       • Independence Check:
         • $P(SD\cap PD)=P(SD)\cdot P(PD)$
         • $0.28=0.4\cdot 0.7$
         • $0.28=0.28$
         • So these are independent events