# 4.4 Random Variables, Density Functions

We flip 3 coins. Let

$C:=\#\text{heads}$ $M:=\begin{cases}1&\text{if all flips match}\\0&\text{otherwise}\end{cases}$ $O:= \text{ odd #heads}$

Let $$I_O$$ be the indicator variable for $$O$$.

We want to show that $$I_O$$ and $$M$$ are independent. Please enter all answers in the form of decimals with three significant digits.

1. What is $$\Pr[I_O=1]$$?

Exercise 1

$$\Pr[I_O=1]=\Pr[O]=\Pr[C=1 \text{OR} C=3]=\Pr[C=1]+\Pr[C=3]=\frac{3}{8}+\frac{1}{8}=\frac{1}{2}$$

2. What is $$\Pr[I_O=0]$$?

Exercise 2

$$\Pr[I_O=0]=1-\Pr[I_O=1]=\frac{1}{2}$$

3. What is $$\Pr[M=1]$$?

Exercise 3

$$M=1$$ if we get all heads or all tails. Hence, $$\Pr[M=1]=\Pr[HHH]+\Pr[TTT]=\frac{1}{8}+\frac{1}{8}=\frac{1}{4}$$.

4. What is $$\Pr[M=1 \text{ AND } I_O=1]$$?

Exercise 4

The only outcome in this event is HHH (all heads).

5. What is $$\Pr[M=0 \text{ AND } I_O=1]$$?

Exercise 5

The event $$[M=0 \text{ AND } I_O=1]$$ is equivalent to $$C=1$$.

6. What is $$\Pr[M=1 \text{ AND } I_O=0]$$?

Exercise 6

The only outcome in this event is TTT (all tails).

7. What is $$\Pr[M=0 \text{ AND } I_O=0]$$?

Exercise 7

The event $$[M=0 \text{ AND } I_O=0]$$ is equivalent to $$C=2$$.

Now you should verify that $$\Pr[M=k_1 \text{ AND } I_O=k_2]=\Pr[M=k_1]\Pr[I_O=k_2]$$ for all $$k_1,k_2\in \{0,1\}$$.