Probability & Statistics Ch.2 Section 3 Question 3

· 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 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 second coin is heads, and Y = 5 if the second coin is tails. Let Z=XYZ = XY. What is the probability function of Z?

Solution

The possible outcomes:

  • First coin tails (X=0): Z = 0 regardless of second coin, probability = 12\frac{1}{2}
  • First coin heads (X=1), second coin heads (Y=1): Z = 1, probability = 12×12=14\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}
  • First coin heads (X=1), second coin tails (Y=5): Z = 5, probability = 12×12=14\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}
pZ(0)=12p_Z(0) = \frac{1}{2}

pZ(1)=14p_Z(1) = \frac{1}{4}

pZ(5)=14p_Z(5) = \frac{1}{4}

pZ(z)=0p_Z(z)=0 for all z∉{0,1,5}z\notin\{0,1,5\}