Probability Distributions   The axiom of risk aversion needs little defense. So far, however, our treatment of risk has been limiting in that it took the variance (or, equivalently, the standard deviation) of port- folio returns as an adequate risk measure. In situations in which variance alone is not ade- quate to measure risk this assumption is potentially restrictive. Here we provide some justification for mean-variance analysis. II. Portfolio Theory 6. Risk and Risk Aversion The McGraw−Hill Companies, 2001           172 PART II Portfolio Theory     The basic question is how one can best describe the uncertainty of portfolio rates of re- turn. In principle, one could list all possible outcomes for the portfolio over a given period. If each outcome results in a payoff such as a dollar profit or rate of return, then this payoff value is the random variable in question. A list assigning a probability to all possible val- ues of a random variable is called the probability distribution of the random variable. The reward for holding a portfolio is typically measured by the expected rate of return across all possible scenarios, which equals   n E(r) Pr(s)r(s) s 1   where s 1, . . . , n are the possible outcomes or scenarios, r(s) is the rate of return for out- come s, and Pr(s) is the probability associated with it. Actually, the expected value or mean is not the only candidate for the central value of a probability distribution. Other candidates are the median and the mode. The median is defined as the outcome value that exceeds the outcome values for half the population and is exceeded by the other half. Whereas the expected rate of return is a weighted average of the outcomes, the weights being the probabilities, the median is based on the rank order of the outcomes and takes into account only the order of the outcome values. The median differs significantly from the mean in cases where the expected value is dominated by extreme values. One example is the income (or wealth) distribution in a pop- ulation. A relatively small number of households command a disproportionate share of to-