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I was taking a look at this book while trying to pick a book for learning some rigorous probability theory. I have been totally stumped by the motivating eg. on the first page.

Specifically, I am confused with the relationship of Z to X and Y. What is the sample space over which Z is defined? Is it {H,T}? What does Z range over?

If Z's domain is {H,T} and range is {X,Y}, then Z isn't even a random variable. As it ranges over two random variables and not real numbers. That is, Z(H)=X and Z(T)=Y.

Is this just a bad example or am I missing something important?

Edit : adding details of how X, Y, and Z are defined.

The book is "A first look at rigorous probability theory by Jeffrey S Rosenthal". I tried linking to books.google.com. The example is on page 1.

X ~ Poisson(5)

Y ~ N(0, 1)

The underlying sample spaces for X and Y are not stated.

X is a discrete random variable while Y is a continuous RV. The author is trying to motivate with his example that the distinction made between discrete and continuous RVs in elementary probability texts is artificial.

The author introduces Z as follows. We let X and Y be as above, and then flip an (independent) fair coin. If the coin comes up heads we set Z = X, while if it comes up tails we set Z = Y. In symbols, P(Z = X) = P(Z = Y) = 1/2. Then what sort of random variable is Z? It is not discrete, since it can take on an uncountable number of different values. But it is not absolutely continuous, since for certain values z (specifically, when z is a non-negative integer) we have P(Z = z) > 0. So how can we study the random variable Z?

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the link you have doesn't always work. Can you put more details of how Z is defined? –  Memming Apr 23 '13 at 15:33
    
I have added more details of how X, Y, and Z are defined. –  user869081 Apr 23 '13 at 15:46

1 Answer 1

up vote 1 down vote accepted

$Z$'s domain is NOT {H,T}. The sample space must represent the joint event of head/tail and also events of X and Y.

$Z$ is a mixed random variable. It is a measurable function from the sample space to reals. Since natural numbers is a subset of reals, this is fine.

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So, assuming X and Y share their sample space (say S) then is the domain of Z - S X {H, T}? Then Z(s,H) = X(s) and Z(s, T) = Y(s), where s belongs to S? Is this understanding correct? –  user869081 Apr 23 '13 at 16:46
    
Sure, that's one possible way to construct a sample space. –  Memming Aug 17 '13 at 0:12

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