【正文】 te expected value, mean, of the rate of return. = [.25 x .08 + .5 x .12 + .25 x .16] = Calculate variance as probabilityweighted squared deviations of values from expected value. ?2 = .25[.08 .12]2 + .5[.12 .12]2 + .25[.16 .12]2 = Finally calculate standard deviation as square root of variance. ? ? = SQRT() = Common Probability Distributions 27 Probability Distributions Probability distribution specifies the probabilities of the possible outes of a random variable. Discrete random variable can take on at most a countable number of possible values, such as coin flip or rolling dice. Continuous random variable can take on an uncountable (infinite) number of possible values, such as asset returns or temperatures. Probability function specifies the probability that the random variable takes on a specific value: P(X = x). Two Key Properties of a Probability Function. 1. 0 ≤ p(x) ≤ 1 because a probability lies between 0 and 1. 2. The sum of probabilities p(x) over all values of X equals 1. 28 Discrete Probability Distributions Discrete Uniform Random Variable: The uniform random variable X takes on a finite number of values, k, and each value has the same probability of occurring, . P(xi) = 1/k for i = 1,2,…,k. Bernoulli random variable is a binary variable that takes on one of two values, usually 1 for success or 0 for failure. Think of a single coin flip as an example of a Bernoulli . Binomial random variable: X ~ B(n, p) is defined is the number of successes in n Bernoulli random trials where p is the probability of success on any one Bernoulli trial. For 4 = X ~ B(10, ) think –“what?s probability of 4 heads in ten coin flips?” Probability distribution for a Binomial random variable is given by: ? ? ? ? xnx ppxnpnxp ??????????? 1,29 Uniform Distribution Population Distribution Summary Measures ??x i i N X N ?? ??????1 2 5 . ????x i x i N X N ?????????? 2 1 1 12 .[ } .0 .1 .2 .3 1 2 3 4 Example of distribution of a uniform random variable. 30 .0 .1 .2 .3 0 2 4 6 8 10 X P(X) Binomial Dist’n – Coin Flips N = 10 p = .5 Probability of exactly 4 heads in 10 tosses = Bar Height Probability of 4 heads or fewer in 10 tosses = Sum of Bar Heights 31 Binomial Probability Dist’n Function P(X= x | n,p) = probability that X = x given n amp。Investment Tools – Probability SASF CFA Quant. Review 2 Probability A random variable is a quantity whose oute is uncertain. Two defining properties of Probability. 1. Probability of any event E is a number between 0 and 1, p(E). 2. Sum of the probabilities of any list of mutually exclusive and exhaustive events equals 1. ? Mutually exclusive = one and only one event can occur at any time. ? Exhaustive = one of the events must occur, jointly cover all possible outes. Empirical probability probability of an event occurring is estimated from data, usually in the form of a relative frequency. A priori probability probability of an event is deduced by reasoning about the structure of the problem itself. Subjective probability probability of an event is based on a personal assessment without reference to any particular data. 3 Visualizing Sample Space 1. Listing S = {Head, Tail} 2. Contingency Table 3. Decision Tree Diagram 4 2 nd Coin 1 st Coin Head Tail Total Head HH HT HH, HT Tail TH TT TH, TT Total HH, TH HT, TT S Contingency Table Experiment: Toss 2 Coins. Note Faces. S = {HH, HT, TH, TT} Sample Space Oute (Count, Total % Shown Usually) Simple Event (Head on 1st Coin) 5 Tree Diagram Oute S = {HH, HT,