【正文】 .2 .4 .6 0 1 2 3 4 5 X P(X) 33 Week 1 Week 2 Week 3 Week 4 S=$100 Assuming: 1. Share Price goes up or down by $10 each week. 2. Probability that Share Price goes down by $10 is 25% each week. . pdown = .25 3. Probability that Share Price goes up by $10 is 75% each week. pup = .75 4. Probability that Share Price goes up this week is NOT affected by what happened before. S=$110 .75 S=$100 .25 P = .75 S=$90 .25 P = .25 S=$120 .75 P = .5625 .75 P = .375 S=$80 .25 P = .0625 S=$130 S=$110 .75 .25 P = .4219 S=$90 .75 .25 P = .4219 S=$70 .75 .25 P = .1406 P = .0156 P = 2(.25)(.75) P = 3(.75)(.75)(.25) Evolution of Share Price 34 Normal Distribution Normal distribution is a continuous, symmetric probability distribution that is pletely described by two parameters: its mean, μ, and its variance, σ2. General Normal random variable – X ~ N(μ, σ2) ? The normal distribution is said to be bellshaped with the mean showing its central location and the variance showing its “spread”. ? A linear bination of two or more Normal random variables is also normally distributed. Standard Normal distribution – Z ~ N(0, 1). ? is a Normal distribution with mean μ=0, and variance σ2=1. 35 Effect of Varying Parameters (?x amp。 ? Symbol (., A ? B) 2. Union Outes in Either Events A or B or Both ?OR? Statement ?x) X f(X) C A B ?A ?C, ?A = ?C ?A = ?B, ?A ?B 36 Normal Distribution General Normal random variable X ~ N(μ, σ2) ? X can be standardized to a Standard Normal random variable. ? Resulting variable has mean zero and variance equal to 1. Calculating probabilities for a normal random variable ? X ~ N(μ, σ2) taking on a range of specified values, say a X b, directly as the area under the normal curve using the cumulative Normal distribution function as: N(a X b| μ, σ2) = N(X b| μ, σ2) N( X a| μ, σ2) . ? You should be able to show what this looks like using a diagram of the Normal distribution. ? ????? XZ37 Confidence Intervals 90% level 95% level 99% level ?x+?x ?x+?x ??x ?X ?x+?x ??x ??x ??x ?x X?= ?x 177。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, TH, TT} Sample Space Experiment: Toss 2 Coins. Note Faces. T H T H T HH HT TH TT H 6 1 2 3 4 5 6 Row Total 1 1/36 1/36 1/36 1/36 1/36 1/36 1/6 2 1/36 1/36 1/36