**WEB–APPENDIX M**

**CONTINUOUS PROBABILITY DISTRIBUTIONS**

*I*n this appendix, we define continuous probability distributions and very briefly describe several that have proven important in financial economics.

## M.1 CONTINUOUS RANDOM VARIABLE

If a random variable can take on any possible value within a range of outcomes, then the probability distribution is said to be a *continuous random variable.* When a random variable is either the price of or the return on a financial asset or an interest rate, the random variable is assumed to be continuous. This means that it is possible to obtain, for example, a price of 95.43231 or 109.34872 and any value in between. In practice, we know that financial assets are not quoted in such a way. Nevertheless, there is no loss in describing the random variable as continuous and in many times treating the return as a continuous random variable means substantial gain in mathematical tractability and convenience.

For a continuous random variable, the calculation of probabilities works substantially different from the discrete case. The reason is that if we want to derive the probability that the realization of the random variable lays within some range (i.e., over a subset or subinterval of the sample space), then we cannot proceed in a similar way as in the discrete case: The number of values in an interval is so large that we cannot just add the probabilities of the single outcomes.

## M.2 PROBABILITY DISTRIBUTION FUNCTION, PROBABILITY DENSITY FUNCTION, AND