STAT 400

Fri. February 7th, 2019

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Independence (cont.)

To recap: two events $A$ and $B$ are independent if and only if $P(A\mid B)=P(A)$, and dependent otherwise.

Important: Just because two events are dependent does not mean that one event was caused by the other; that is to say, dependence is not the same as causality. In statistics, we only study the correlation between events.

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Properties of Independence

1. Independence is symmetric: $P(A\mid B)=P(A)\leftrightarrow P(B\mid A)=P(B)$.
2. If $A$ and $B$ are independent, then $A'$ and $B$ are also independent:
   $\begin{aligned} P(A'\mid B)&=\frac{P(B\cap A')}{P(B)}\\&=\frac{P(B)-P(B\cap A)}{P(B)} \\&=1-\frac{P(B\cap A)}{P(B)}\\&=1-P(A\mid B)\\&=1-P(A)=P(A'). \end{aligned}$
3. $A$ and $B$ are independent if and only if $P(A\cap B)=P(A)\cdot P(B)$.

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Ex. Toss a coin ten times.
$\begin{aligned} A&=\{\text{first 9 tosses are all H}\}\\ B&=\{\text{10th toss is T}\}\\ C&=\{\text{at least one T within the first 9 tosses}\} \end{aligned}$

Intuitively:

• $A$ and $B$ are independent: they care about different tosses
• $A$ and $C$ are dependent: $C=A'$, so they are disjoint and disjoint events are always dependent
• $B$ and $C$ are independent: they also care about different tosses

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Independence of more than two events

Definition. Given events $A_1,A_2...A_n$, we say they are mutually independent if, for every $k=2,3,...n$ and every subset of indices $i_1,i_2...i_k$, $P(A_{i_1}\cap A_{i_2}\cap A_{i_3}...\cap A_{i_k})=P(A_{i_1})\cdot P(A_{i_2})...P(A_{i_k}).$

For instance, if we want to determine if $A_1, A_2, A_3$ are mutually independent, then all of the following equations must hold:

• $k=2$: 3 equations
  • $P(A_1\cap A_2)=P(A_1)\cdot P(A_2)$
  • $P(A_1\cap A_3)=P(A_1)\cdot P(A_3)$
  • $P(A_2\cap A_3)=P(A_2)\cdot P(A_3)$
• $k=3$: 1 equation
  • $P(A_1\cap A_2\cap A_3)=P(A_1)\cdot P(A_2)\cdot P(A_3)$

The number of equations needed to check grows exponentially, so you probably won’t be asked to check more than three events.

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Ex. Roll a die twice. $\begin{aligned} A_1&=\{\text{first roll is 3}\}\\ A_2&=\{\text{second roll is 4}\}\\ A_3&=\{\text{sum of the two rolls is 7}\} \end{aligned}$ Are these events mutually independent?

• $k=2$:
  • $\frac{1}{36}=P(A_1\cap A_2)=P(A_1)\cdot P(A_2)=\frac{1}{6}\cdot \frac{1}{6}$
  • $\frac{1}{36}=P(A_1\cap A_3)=P(A_1)\cdot P(A_3)=\frac{1}{6}\cdot \frac{1}{6}$
  • $\frac{1}{36}=P(A_2\cap A_3)=P(A_2)\cdot P(A_3)=\frac{1}{6}\cdot \frac{1}{6}$
• $k=3$:
  • $\frac{1}{36}=P(A_1\cap A_2\cap A_3)\ne P(A_1)\cdot P(A_2)\cdot P(A_3)=\frac{1}{6}\cdot \frac{1}{6}\cdot \frac{1}{6}$

Because all of the $k=2$ cases are true, we call these events pairwise independent. However, the $k=3$ case is false, so they are not mutually independent.

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Random Variables

We have already studied the probability of random outcomes occurring quantitatively, but we also want to be able to quantitatively study the outcomes themselves. Often these already have a numerical value associated with them (like a die roll), but sometimes they don’t (like a coin flip). In these cases we can create our own mapping to numerical values, for instance: $\{H,T\}\rightarrow \{1,0\}.$

Definition. Given a sample space of some experiment, a random variable is a rule that associates a value with each outcome in $S$; that is, a random variable $X$ is a mapping from $S$ to $\R$. $X: S \rightarrow \R.$