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### 12 Cards in this Set

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 What is the formula for a conditional probability? P(A│B) = P(AB)/P(B), P(B)≠0. "│" signifies "given that". What is the probability of an apple falling given that Billy shook the tree. P(A) = Apple Falling P(B) = Billy shook the tree P(AB) = P(A│B)*P(B) for a conditional probability. P(A)*P(B) for an independent probability. However, a conditional probability doesn't have independent variables - obviously, so P(A)*P(B) doesn't apply to this problem. What is the addition rule for probabilities and what is its significance? P(A OR B)= P(A) + P(B) - P(AB). Think of a Venn Diagram where the intersecting portion needs to be subtracted out. It's the same formula for P of A OR B. What is the definition of independent event (formula)? What is the multiplication rule for independent events? Definition of independent events:Two events A and B are independent if and only if: P(A│B) = P(A) or P(B│A) = P(A) Multiplication rule for independent event: When 2 events are independent, the joint probability [P(AB)] of A and B equals the product of the individual probabilities A and B. THIS IS ONLY TRUE FOR INDEPENDENT EVENTS. P(AB) = P(A)P(B) What is the total probability rule and formula? The total probability rule explains the unconditional probability of the event in terms of probabilities conditional on the scenarios: a) P(A) = P(AB) + P(AB') P(A) = P(A│B)P(B) + P(A│B')P(B') Probability of A occuring is equal to the probability of A and B occuring and the probability of A occuring and B not occuring (or all of A without the intersection of A and B)where it's a Venn Diagram. b) P(A) = P(AB1) + P(AB2) + ... + P(ABn) P(A) = P(A│B1)P(B1) + P(A│B2)P(B2) + ...+ P(A│Bn)P(Bn) B = Scenario Happens B' = Scenario doesn't happen How do you change a probability into odds and vice versa? As an example assume: P(E) = 1/8 or .125 i.e. probability of even E happening is 1/8. What is another name for marginal probability? If the probability is 1/8, it means that 1/8 of a total would have that outcome, to convert: use this formula --> P(E)/(1-P(E)) or 1/8 / 7/8 = 1/8 * 8/7 = 8/56 = 1/7 Which translates to for every outcome E there are 7 outcomes other than E. To convert backwards just add the numerator to the denominator ie 1/(1+7)= 1/8 Marginal probabilities are also referred to as unconditional probabilities or probabilities not contingent on other factors. What is the difference between mutually exclusive and exhaustive? What is an empirical probability? What is a piori probability? The term mutually exclusive means that only one event can occur at one time; exhaustive means that the events cover all possible outcomes; If exhaustive all probabilities add up to 1. An empirical probability is based on historical data and is most common in the investment industry. Priori probability is based on logical analysis rather than on observation or personal judgement. What is an event? What kind of variable is the return on a risky asset? An event is a specified set of outcomes e.g. the portfolio earns less than 10% is one possible event. The return on a risky asset is an example of a random variable, a quantity whose outcome is uncertain. What is the formula for expected value and give an example? E(X)= Probability weighted average of X e.g. Prob EPS .2 1.00 .3 2.00 .5 3.00 E(EPS) = (.2)(1) + (.3)(2) + (.5)(3) = .2 + .6 + 1.5 E(EPS)= 2.3 What is the formula for variance of a portfolio and an example? The variance of a random variable is the expected value of squared deviations from the random variable's expected value. Variance = P(X1)[x - E(X)]^2 Example Probability EPS .2 1 .3 2 .5 3 So Variance (EPS) = .2(1 - 2.3)^₂ + .3(2-2.3)^₂ + .5(3-2.3)^₂ What is the definition of covariance? What is the formula for covariance? Given 2 random variables R1 and R2, the covariance between R1 and R2 is: COV(R1,R2) = ∑ P(R1,R2)[(R1 - E(R1))(R2 - E(R2)] The deviations should always be written in whole numbers while the probabilities at the beginning should be in decimal form. What formula and how would you solve the following problem: If you believe the probability of getting a positive trading error on your portfolio compared with EAFE to be 90%, what is the likelihood of you being correct if in only 6 out of 8 times you had a positive tracking error? First, you should know that a tracking error is the difference between your return and some benchmark. Second you should use the following formula, which is used for finding the likelihood of an event given a known probability of each occurence and a sample set of data: N!/[(N-X)! * X!] * P^X(1-P)^(N-X) 8!/[(8-6)! * 6!] * .9^6(.1)^2 27*(.53144)(.01)= .1488 or 14.3% and then you have to do the same for 5,4,3,2,1,0, which are the following answers respectively: 5 ==>.0331 4 ==>.0046 3 ==>.0004 2 ==>.0000 1 ==>.0000 0 ==>.0000 Therefore, there is a 18.69% chance of getting 6 out of 8 tracking errors within the range and still have it be 90% likelihood in the long run. Alternatively you could figure out the likelihood of 7 and 8 and subtract that from 1. How do you find the variance of a Bernoulli variable? What the formula and an example? First you have to recognize what a Bernoulli variable is, it's a variable that has one of two outcomes: e.g. on or off, 0 or 1, passed or failed. Suppose you're aksed to estimate the # of bonds expected to default out of 25 bonds. If the default rate is 10.7%, then (1) What is the expected # of defaults? (2) Estimate the variance of the # of defaults? (1) N=25, P=.107; NP=Mean or Expected Value ==> 2.675 or approximately 3. (2) Variance = NP(1-P)=2.675(.893)==>2.38877 The trick in this problem is recognizing that this is a Binomial/Bernoulli variable AND that every observance is independent of the other. That is one bond defaulting is not correlated with another, which of course we know not to be true because of economic factors affecting all bonds. Lastly, notice there is only on probability. If there had been multiple probabilities then we would have used a prior formula.