Discrete random variable

CFA Level I
#1

I don’t understand this statement:

Because we can count all the possible outcomes of X and Y (even if we go on forever in the case of Y),

(Institute 517)

Institute, CFA. 2016 CFA Level I Volume 1 Ethical and Professional Standards and Quantitative Methods. CFA Institute, 07/2015. VitalBook file.

The citation provided is a guideline. Please check each citation for accuracy before use.

How can you count something that goes on forever? What exactly is meant by count? I’m a bit confused why a discrete random variable can go on forever and still be discrete?

#2

You have to look at the entire sentence:

For example, a discrete random variable X can take on a limited number of outcomes x1, x2, …, xn (n possible outcomes), or a discrete random variable Y can take on an unlimited number of outcomes y1, y2, … (without end).1 Because we can count all the possible outcomes of X and_Y_ (even if we go on forever in the case of Y), both X and Y satisfy the definition of a discrete random variable

They are comparing two types of discrete random variables - ones with finite possibilities (e.g., 1, 2, or 3) and discrete random variables with infinite possibilities (e.g., 1,2,3,4,… all the way up). For each possibility, there is a probability attached to each possible value coming up, which may be 0.

This is distinct from continuous random variables which can be any number between a given range (e.g., pick any real number between 0 and 1 or pick any real number between +/- infinity). With a continuous random variable, the probability of any particular result by itself is 0. (e.g., the value will never be exactly 1.0000000000…). You have to look at the probability of a range (e.g., the probability of the result falling between .25 and .75).

Hope this helps.

#3

Often, people in finance use technical language that they don’t understand fully. The problem is that by not understanding it fully, they’ll end up making stupid statements.

Such is the case here, though not for the reason you’d suspect.

In mathematics, we describe a set as countable if its members can be put into a one-to-one (1-1) correspondence with a subset of the whole numbers (positive integers). Thus, we would describe the number of players on a baseball team as countable because we can create the following 1-1 correspondence:

  1. Pitcher
  2. Catcher
  3. First baseman
  4. Second baseman
  5. Third baseman
  6. Shortstop
  7. Left fielder
  8. Center fielder
  9. Right fielder

The catch – and this is _ not _ where CFA Institute erred – is that this definition of countable means that some infitite sets are countable. As a trivial example, the set of whole numbers is countable, even though it is infinite:

  1. 1
  2. 2
  3. 3
  4. 4
  5. 5

and so on.

The place where CFA Institute erred is in stating (or, at least, implying) that if a set of outcomes is countable, then it necessarily represents a discrete random variable.

That’s simply not true.

For example, the set of all positive rational numbers is countable (and the proof of this is very clever), but a random variable that can take on any positive rational value would likely not be described as discrete. (This is a touchy area, and gets into measure theory, and the fact that the rationals are not dense in the reals, and so on; far too technical to get into here.)

The fact remains that some infinite sets are countable. Don’t fight it: accept it and move on.

#4

An interesting thought exercise is to think about a discrete random distribution with infinite non-continuous, but countable possibilities within a finite range (e.g., any rational number between 0 and 1). Functionall, it is like a uniform continuous distribution with an infinite number of discontinuous points between each possible value. Basically, its like trying to integrate the salt and pepper function.

#5

Thanks for a disturbing flashback to grad school.

#6

By this time you shouldn’t need a warning on my posts. You read them at your own peril.

#7

Help! I’m being repressed!

#8

simple example, you are made bald and asked to count your hair, doesn’t matter it may take you hours or days to count it, however, 250,451,001 may be a possible answer (may be inflated) though it may seem to go on and on, you can still count it…

#9

A lot of hair is countable, not going to argue that one, I was referring to an infinite number of values.

#10

Thanks for the indepth response, It’ll take me some time to get my head around what you described.

So basically if we have whole numbers, then they are discrete, if we have rational/irrational numbers then they are not?

#11

It doesn’t have to be whole numbers per se, but it has to attach a single probability to each possible value. For example, you could have fractions (rational numbers): 50% chance of .10 return and 50% chance of .15 return. To be honest with you, if you are just trying to nail this down for the exam, I don’t think I would worry too much about the discrete distributions with inifinite countable possibilties. Though I defer to those who have actually taken the test, that seems a little high octane for what they are looking to test.

#12

Exactly. Forest for the trees.