Probability & Statistics Ch.2 Section 3 Question 2

· Mohammad-Ali Bandzar

Solutions for “Probability and Statistics: The Science of Uncertainty” (Second Edition). These are solutions I have come up with; I offer no guarantee of accuracy.

Question

Consider flipping a fair coin. Let Z = 1 if the coin is heads, and Z = 3 if the coin is tails. Let W=Z2+ZW = Z^2 + Z.

a) What is the probability function of Z? b) What is the probability function of W?

Solution

(a) What is the probability function of Z?

pZ(1)=12p_Z(1) = \frac{1}{2}

pZ(3)=12p_Z(3) = \frac{1}{2}

pZ(z)=0p_Z(z)=0 for all z∉{1,3}z\notin\{1,3\}

(b) What is the probability function of W?

When Z = 1: W=12+1=2W = 1^2 + 1 = 2 When Z = 3: W=32+3=12W = 3^2 + 3 = 12

pW(2)=pZ(1)=12p_W(2) = p_Z(1) = \frac{1}{2}

pW(12)=pZ(3)=12p_W(12) = p_Z(3) = \frac{1}{2}

pW(w)=0p_W(w)=0 for all w∉{2,12}w\notin\{2,12\}