BT IS A GREAT BUY!
27 Aug 2008
Apr
04
The difference between marginal, conditional and joint probabilities
by David Harper, CFA, FRM, CIPM
In finance, we are careful to distinguish between marginal, conditional and joint probabilities (for the FRM candidate, these terms are reviewed in Chapter Two of Gujarati's Essentials of Econometrics). To illustrate, let's assume we have an operational process like widget production. As the process is not perfect, errors sometimes happen, and the errors are either technical or human. Over the last 70 days, we observe the errors. On any given day, there occurred zero to three (0 to 3) human errors and zero to three (0 to 3) technical errors. The frequency distribution looks like this:
Joint probability
Each cell contains a joint probability. If 10 out of the 70 days were days that saw three human errors and three technical errors, then the joint probability is P(Human Error =3, Technical Error = 3) = 10/70 = 14.3% (rounded below to .14)
Note the interior cells sum to 1.0 as these are discrete variables (one way to remember this: if we can count the errors, they are discrete. If we must measure the errors on a scale, they are likely continuous) and they are mutually exclusive (you can't have both 1 and 2 errors on the same day, it's either 0 or 1 or 2 or 3) and cumulatively exhaustive (0 to 3 captures all of the possibilities).
Marginal probability
The marginal probability is the unconditional probability. It is the probability of a human or technical error rate without any prior information (unconditional on any information). In the case, for example, the unconditional probability of one (1) Human error is the sum of a column: P(Human Error =1) = (0.06 + 0.09 + 0.04 + 0.01) = (0.2).
Both the joint probability and the marginal probability lack prior information, they are unconditional. One is the unconditional probability of two outcomes; e.g., P(Human = x, Technical = y). The other is the unconditional probability of one outcome; e.g., P(Human = x), P(Technical = y).
Conditional probability
A conditional probability is typically more realistic. Conditional means "we have some prior information." Assume we already know (or expect) zero human errors for a given day. Without prior information, we understood the unconditional probability of zero technical errors to be 20% (see end of first row above). But instead we know that zero technical errors occurred (the orange column below). The probability of zero technical errors given zero human errors is about 55% (0.09/0.16. Rounding creates some error).
Notice that the prior knowledge of zero human errors changes the probability of zero technical errors from an unconditional 20% to a conditional 55%. We express this as: P (Technical Errors = 0 | Human Errors = 0) = 9%/16%. Where the pipe "|" means "given that" or "conditional on."
Just three applications (regression, volatility, copulas)
As Gujarati shows nicely, a linear regression can be viewed as a conditional linear regression. If the line is, say, Y = mX + b, then we can say E (Y | X ) = mX + b. In words, given a value of X (prior information that X = x), we expect a value of Y that is a function of X.
Any of the useful volatilities approaches are about conditional volatility; e.g., moving average, exponentially weighted moving average (EWMA) and GARCH(1,1). We are trying to estimate current variance conditional on prior information (yesterday's variance).
In modeling the default correlation among, say, a basket of credit obligors, it is common to use a copula function. A copula is a joint probability. Where F(x) is the marginal probability that a credit obligor will default, and F(y) is the marginal probability that different credit obligor will default, we are interested in the joint probability that both default, P (F(x) < default threshold and F(y) < default threshold). The copula function translates the two marginal probabilities into a single joint probability.
Comments
Be the first to leave a comment!
Leave a Comment