May 14

A grid for grasping probability distributions

by David Harper, CFA, FRM, CIPM

FRM | Risk | Quant | |

For the FRM, you might be new or rusty to the statistics section. Before you launch into random variables, put the distribution list into perspective by realizing they all fall into the 2x2 grid below. What is a probability distribution? It is a mathematical curve that describes a random variable; it illustrates the possible outcomes for the random variable.

The two key dimensions are:

  • discrete versus continuous, and
  • local (density) versus Cumulative

 

 

Discrete random variables includes dice (1,2,3,...), currencies, and tests like the FRM which are pass/fail but nothing in between. Continuous random variables can be measured but they cannot be counted because there is always some number in between two others; e.g., time and distance are good examples.

A probability density function (pdf) is denoted with a lower-case function as in P(X=x) = f(x). It shows the "local" probability that a random variable will be equal to some value, or in the case of a continuous variable, within some interval. For a single six-sided die, a pdf function would be P(X=3)=f(x)=1/6.

A cumulative distribution function is denoted with an upper-case function as in P(X<=x)=F(x). It shows the cumulative probability that a random variable will be less than or equal to some value. For a dice, a cumulative function would be P(X<=3)=F(X)=3/6.

In risk measurement, a typical situation is to assume the normal distribution for a random variable like asset return (knowing normal tails are too skinny). The normal is continuous, unlike the binomial and the Poisson which are discrete. Then we often want to know about the worst 1% or 5% of outcomes. That is asking about the area under the curve in the tail of the p.d.f. that occupies 1% or 5% of the total area under the p.d.f. curve. That area maps directly to a point on the cumulative curve. So, a typical risk question is about the 1% or 5% cumulative function.

Comments

  1. John Vesey 16 May 2007

    David,
    I attended a session with Robert Arnott, during which he outlined his RA Fundamental Index Fund approach. Subsequently, I have accessed his Reasearch Affiliates web site and read some their papers. Have you studied these? If so, what are your thoughts as to their tangential investment approach to CAPM criteria?
    As always, your articles, such as the one above, are excellent--keep up the great work---JCV

  2. Mohammad al-Alwan 24 May 2007

    i really like your excellent way of explaining these mathematical concepts and i wonder if you have same notes structure for CFA

  3. John Vesey 24 May 2007

    David,

    Schwab did a teleconference with Arnott this afternoon as they are the underwriters of RA’s Fundamentals Index product. If interested, I shall send you one of their pro-fromas as well as Arnott’s analysis. (Even though I still have my NASD licenses, I am not sellling this or any other fund to 3rd parties)

    Always great to access your site, Mr. Harper--just outdstanding work. On a lighter note, I acquired a mathematical CD program for my grandchildren titled “The Joy of Mathematics.” It is done by the Teaching Company and the instructor is a prof at Harvey Mudd. He does a wonderful job with math and his video on Fibonacci to calculus is terrific--almost as good as you are!!!

  4. jshashank 19 Jun 2007

    Great David,That was great refresher for me!

  5. jcvesey@sbcglobal.net 31 Aug 2007

    David,
    I am always pleased to see reference to you via someone else’s work. While reviewing James Hitchener’s “Financial Valuation” book, I came across a footnote on page 168 which acknowledged your contribution via Investopedia on the Gordon Growth Model. Have a wonderful Labor Day weekend. John Vesey

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