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Definition
-one whose values or results are variable
--> ex) when you roll a die, the number you get is a random variable
-probability of one means that the event is not random |
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-single outcome or a set of outcomes
--> ex) roll a die, event that you get a 1; another is getting even numbers |
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| Mutually Exclusive Events |
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Definition
| -events that cannot both happen at the same time. the occurrence of one precludes the occurrence of the other |
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| -cover the range of all possible outcomes of an event |
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Definition
| -observed value of a random variable |
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| Properties of Probability |
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Definition
-probability of any event is between 0 and 1
-For a set of events that are mutually exclusive and exhaustive, the sum of probabilities equals 1 |
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Definition
-calculated by analyzing PAST data
-estimates the probability of an event based on its frequency of occurrence in the past |
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Definition
| -calculated by using formal analysis and reasoning |
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Definition
-draws on subjective reasoning and personal judgement to estimate the probability of an event
--> ex) estimate a probability of stock increase bc of new CEO that you believe to be really good |
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Definition
-states as the probability of the event occurring to the probably of the event not occurring P(E) to [1-P(E)]
-P(E)= a / (a+b) |
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Definition
-stated as the probability of an event NOT occurring to the probability of the event occurring [1-P(E)] to P(E)
-P(E) = b / (a+b) |
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Definition
-the probability of an event regardless of the outcomes of other events
-ex) What is the probability of a return on a stock above 10%? |
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Term
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Definition
-the probability of an event occurring given that another event has occurred
-calculated using joint probabilities
ex) What is the probably of the return on a stock being more than 10% GIVEN that the return is above the risk free rate? |
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Term
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Definition
-the occurrence of one is related to the occurrence of the other
--> ex) prob of doing well on an exam is related to probability of preparing well for it |
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Term
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Definition
-occurrence of one event does not influence the occurrence of the other
-P(AlB) = P(A) and vice versa
--> ex) knowing probability it will rain tomorrow will tell nothing about probability of stock mkt going up |
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Term
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Definition
-P(AB) = P(AlB) * P(B)
-the probability that both of two events will occur
--> ex) what is the probability of BOTH A and B happening?
-If A is contained within the set of possible outcomes for B, P(AB) = P(B) |
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Term
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Definition
-Joint probability of two independent events equals the product of the individual probabilities
-P(AB) = P(A) * P(B) bc P(AIB) = P(A) bc A does not rely on B |
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Term
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Definition
-P(A or B) = P(A) + P(B) - P(AB)
-probability of A happening or B happening OR both happening |
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Definition
P(A) = P(AlB) * P(B) + P(AlB^c) * P(B^c)
P(B^c) is the probability of B NOT occurring
-expresses unconditional probably in terms of conditional probabilities for mutually exclusive and exhaustive events
-P(A) is a weighted average in TPR |
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Term
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Definition
-P(AlB)= P(A) * {P(BlA)/P(B)}
-allows us to adjust our viewpoint when we receive new information
-most likely have to use TPR to get unconditional probability |
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Term
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Definition
| the probability of a specific stock value (furthest out on tree diagram) is the probability that the economy is good AND the probability that the stock is that price |
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| Expected Value of a Random Variable |
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Definition
| -the probability weighted average of all possible outcomes for the random variable |
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Term
| Variance of a Random Variable |
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Definition
-refer to equation sheet
-no units
-if ZERO, there is no dispersion in distribution so outcome is certain and variable is NOT a random variable |
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| Standard deviation of random variable |
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Definition
-positive square root of the variance
-same unit as random variable |
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Term
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Definition
-measure of the extent to which two random variables move together
-->like variance, except that variance measures how a random variable varies with itself
-Covariance is symmetric Cov (X,Y) = Cov (Y,X)
-values range from minus infinity to positive infinity
--> minus means variables move in opposite directions, when return on one asset is above its EV, the return on the other tends to be below its EV
-Cov (X,X) = Var (X)
-Cov=0 if terms are unrelated
-refer to equation sheet |
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Term
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Definition
-difficult to interpret units
-can take on extreme large values
-does not say anything about strength of the relationship btwn variables |
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Definition
-STANDARDIZED measure of the LINEAR relationship between two variables
-measures the strength and direction of relationship
-no unit
-lies between -1 and +1
- Corr=+1 indicates perfectly positive correlation
- Corr= -1 indicates perfectly negative correlation
-Corr = 0 indicates no linear relationship |
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Term
| Multiplication Rule of Counting |
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Definition
| If one event can occur in ''m'' ways and another event can occur in ''n'' ways, then the number of ways that ''both'' events can occur together is ''m * n' |
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Term
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Definition
| -Given n items, there are n! ways of arranging them |
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Term
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Definition
-n items of which each can receive one of k labels. The number of items that receive label A is n1 and the # that receive label B is n2 so:
n1+n2+...nk = n
# of ways in which labels can be assigned =
n! / [(n1!)*(n2!)*...(nk!)] |
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Term
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Definition
nCr = n! / (n-r)!*r!
-Number of ways to choose r objects from total of n objects
-the order or rank of labeling is NOT important |
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Term
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Definition
nPr = n! / (n-r)!
-Number of ways to choose r objects from a total of n objects when the order in which the r objects are chosen DOES matter |
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Term
| conditional expectation in investment application |
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Definition
-E(X) = E(XlS1)*P(S1)+E(XlS2)*P(S2)+...E(XlSn)*P(Sn)
-goal is to calculate the expected value of a random variable given all the possible scenarios that can occur
-similar to total probability rule which states unconditional probabilities in terms of conditional probabilities |
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Term
| Expected Value (of returns on a portfolio) |
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Definition
-function of the returns on the individual assets and of their respective weights
-refer to equation sheet |
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Term
| Weights of individual assets in portfolio |
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Definition
| Weight of asset i = Market value of investment i / market value of portfolio |
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Term
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Definition
-function of individual asset weights and variances AND the covariances of the assets with each other
-to calculate variance for a portfolio containing n different assets we would require n(n-1)/2 covariances |
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Term
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Definition
| - X and Y are uncorrelated |
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