Chapter 2 Section 3 Question 4
Question
Consider flipping two fair coins. Let $X = 1$ if the first coin is heads, and $X = 0$ if the first coin is tails. Let $Y = 1$ if the two coins show the same thing (i.e., both heads or both tails), with $Y = 0$ otherwise. Let $Z = X + Y$, and $W = XY$.
  1. What is the probability function of Z?
  2. What is the probability function of W?
Answer
Author: Mohammad-Ali Bandzar| Date:Oct 15 2020
(a) What is the probability function of Z?
valid outcomes for z=0
Tails,Heads $$p\small{Z}(0) = \frac{1}{2}*\frac{1}{2}=\frac{1}{4}$$ valid outcomes for z=1
Heads,Tails
Tails,Tails $$p\small{Z}(1) = \frac{1}{2}*\frac{1}{2}+\frac{1}{2}*\frac{1}{2}=\frac{1}{2}$$ valid outcomes for z=2
Heads,Heads $$p\small{Z}(2) = \frac{1}{2}*\frac{1}{2}=\frac{1}{4}$$ Otherwise:
$p\scriptsize{Z}\normalsize{(z)=0}$ for all $z\notin\{0,1,2\}$
(b) What is the probability function of W?
valid outcomes for w=0
Heads,Tails
Tails,Tails
Tails,Heads $$p\small{Z}(0) = \frac{1}{2}*\frac{1}{2}+\frac{1}{2}*\frac{1}{2}+\frac{1}{2}*\frac{1}{2}=\frac{3}{4}$$ valid outcomes for w=1
Heads,Heads
$$p\small{Z}(1) = \frac{1}{2}*\frac{1}{2}=\frac{1}{4}$$ Otherwise:
$p\scriptsize{Z}\normalsize{(z)=0}$ for all $z\notin\{0,1\}$