**WEB-APPENDIX L**

**SOME ELEMENTARY CONCEPTS INVOLVING PROBABILITY**

*I*n this appendix, we summarize the probability concepts we rely on throughout this book. We present them in the context of discrete probability distributions, for example:

*r*_{p} |
*Probability* |

−0.10 |
0.20 |

0.00 |
0.40 |

0.10 |
0.40 |

## L.1 PROPERTIES OF THE EXPECTED VALUE OF A RANDOM VARIABLE

We first look at three useful properties of expected values:

*Property* 1: |
The expected value of a constant is that constant. In particular, we shall make considerable use of the notion throughout the book of a risk-free rate of return *r*_{f}; thus, *E*(*r*_{f}) = *r*_{f}. |

*Property* 2: |
The expected value of a constant times a random variable is the constant times the expected value of that random variable. In particular, if *x*_{i} is the proportion of a portfolio invested in security *i* and *r*_{i} is its return, then *E*(*x*_{i}r_{i}) = *x*_{i}E (*r*_{i}). |

*Property* 3: |
The expected value of a sum equals the sum of the expected values. In particular, suppose a portfolio consists of two securities, 1 and 2; then *E*(*r*_{p}) = *E*(*x*_{1}*r*_{1}) + *E*(*x*_{2}*r*_{2}) = *x*_{1}*E*(*r*_{1}) + *x*_{2}*E*(*r*_{2}). This property is quite general and applies also to sums of functions of random variables. |

## L.2 MEASURES OF DISPERSION FOR RANDOM VARIABLES

We next look at measures of dispersion for random variables, defining first the variance,

Since the term *E*(*r*) inside the braces can be treated as a constant,

We then define standard ...