Chapter 3 problem 40a - potentially wrong answer

[Greg-Mos]

Let X be the r.v. that maps Tom's answer to a unique (for each answer) integer. Let Y be the r.v. that maps my answer to a unique (for each answer) integer.

"I trust Tom, so I will always go with Tom’s answer". So, Y = g(X), where g is the identity function, which is a one-to-one function.

Therefore, according to the discussion on page 125, following Definition 3.7.1, X and Y have the same PMF:

x	P(X = x)	y	P(Y = y)
x1	p1	g(x1) = x1	p1
x2	p2	g(x2) = x2	p2
x3	p3	g(x3) = x3	p3
:	:	:	:

Alternatively, consider a board game where a player rolls a 4-sided die and moves as many squares forward as the number on the die. Let W be the result of the die roll and Z be the number of squares the player moves forward. From the description, P(W=Z) = 1 and Z = g(W), where g is the identity function, which is one-to-one. Therefore, W and Z have the same PMF :

w	P(W = w)	z	P(Z = z)
w1	p1	g(w1) = w1	p1
w2	p2	g(w2) = w2	p2
w3	p3	g(w3) = w3	p3
w4	p4	g(w4) = w4	p4