** 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 |

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; thus, _{f}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 is the proportion of a portfolio invested in security _{i}i and r is its return, then _{i}E(x) = _{i}r_{i}x (_{i}Er)._{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. |

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 ...

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