8. PROBABILITY CONCEPTS
A. Define a random variable, an outcome, an event, mutually exclusive events, and exhaustive events :
- A random variable is an uncertain quantity/number.
- An outcome is an observed value of a random variable.
- An event is a single outcome or a set of outcomes.
- Mutually exclusive events are events that cannot both happen at the same time.
- Exhaustive events are those that include all possible outcomes.
B. State the two defining properties of probability and distinguish among empirical, subjective, and a priori probabilities:
There are two defining properties of probability :
- The probability of occurrence of any event (Ei) is between 0 and 1
- If a set of events E1, E2, …, En, is mutually exclusive and exhaustive, the probabilities of those events sum up to 1.
An empirical probability is established by analyzing past data.
An a priori probability is determined using a formal reasoning and inspection process.
Empirical and a priori probabilities are objective probabilities.
A subjective probability is the least formal method of developing probabilities and involves the use of personal judgment.
C. State the probability of an event in terms of odds for and against the event :
Stating the odds that an event will or will not occur is an alternative way of expressing probabilities. Consider an event that has a probability of occurrence of 0,125, which is one-eighth. The odds that the event will occur are 0,125/(1 – 0.125) = 1/7, which we state as, “the odds for the event occurring are one-to-seven”. The odds against the event occurring are the reciprocal of 1/7, which is seven-to-one.
D. Distinguish between unconditional and conditional probabilities :
Unconditional probability refers to the probability of an event regardless of the past or future occurrence of other events.
A conditional probability is one where the occurrence of one event affects the probability of occurrence of another event. Using probability notation, “the probability of A given the occurrence of B” is expressed as P(A/B), where the vertical bar (/) indicates “given”, or “conditional upon”. A conditional probability of an occurrence is also called likelihood.
E. Explain the multiplication, addition, and total probability rules :
The multiplication rule of probability is used to determine the joint probability of two events :
P(AB) = P(A/B) * P(B)
The addition rule of probability is used to determine the probability that at least one of two events will occur :
P(A or B) = P(A) + P(B) – P(AB)
The total probability rule is used to determine the unconditional probability of an event, given conditional probabilities :
P(A) = P(A/B1) * P(B1) + P(A/B2) * P(B2) +…+ P(A/Bn) * P(Bn)
where B1, B2, …, Bn is a mutually exclusive and exhaustive set of outcomes.
F. Calculate and interpret 1) the joint probability of two events, 2) the probability of at least one of two events will occur, given the probability of each and the joint probability of the two events, and 3) a joint probability of any number of independent events :
The joint probability of two events is the probability that they will both occur : P(AB) = P(A/B) * P(B).
Calculating the Probability That at Least One Event Will Occur
The addition rule for probabilities is used to determine the probability that at least one of two events will occur : P(A or B) = P(A) + P(B) – P(AB).
For mutually exclusive events, where the joint probability P(AB) = 0, the probability that either A or B will occur is simply the sum of the unconditional probabilities for each event : P(A or B) = P(A) + P(B).
Calculating a Joint Probability of any Number of Independent Events
When dealing with independent events, the word “and” indicates multiplication, and the word “or” indicates addition.
P(A or B) = P(A) + P(B) – P(AB), and P(A and B) = P(A) * P(B)
G. Distinguish between dependent and independent events :
Independent events refer to events for which the occurrence of one has no influence on the occurrence of the others. The definition of independent events can be expressed in terms of conditional probabilities. Events A and B are independent if and only if P(A/B) = P(A), or equivalently P(B/A) = P(B).
H. Calculate and interpret an unconditional probability using the total probability rule :
The total probability rule highlights the relationship between unconditional and conditional probabilities of mutually exclusive and exhaustive events. It is used to explain the unconditional probability of an event in terms of probabilities that are conditional upon other events.
Expected Value
The average value for a random variable that results from multiple experiments is called an expected value.
The expected value is the weighted average of the possible outcomes of a random variable, where the weights are the probabilities that the outcomes will occur. The mathematical representation for the expected value of random variable X is :
E(X) = P(X1) * x1 + P(X2) * x2 + … + P(Xn) * Xn
The expected value is, statistically speaking, our best guess of the outcome of a random variable.
The variance is calculated as the probability-weighted sum of the squared differences between each possible outcome and the expected value.
I.Explain the use of conditional expectation in investment applications :
Expected values can be calculated using conditional probabilities. As the name implies, conditional expected values are contingent upon the outcome of some other event. An analyst would use a conditional expected value to revise his expectations when new information arrives.
J. Explain the use of a tree diagram to represent an investment problem :
A general framework called a tree diagram is used to show the probabilities of various outcomes.
K. Calculate and interpret covariance and correlation :
The variance and standard deviation measure the dispersion, or volatility, of only one variable. In many finance situations, however, we are interested in how two random variables move in relation to each other (return of two assets).
Covariance is a measure of how two assets move together. It is the expected value of the product of the deviations of the two random variables from their respective expected values :
Cov(Ri,Rj) = E{ (Ri – E(Ri)) * (Rj – E(Rj)) }
The following are properties of covariance :
- The covariance is a general representation of the same concept as the variance. That is, the variance measures how a random variable moves with itself, and covariance measures how one random variable moves with another random variable.
- The covariance of RA with itself is equal to the variance of RA ; that is Cov(RA,RA) = Var(RA).
- The covariance may range from negative infinity to positive infinity.
Basically :
- If covariance is negative, the two returns move in opposite directions.
- If covariance is positive, the two returns move in the same direction
- If covariance is zero, the one return doesn’t move, regardless of movement in the other return.
The correlation coefficient, or simply, correlation, is :
Properties of correlation of two random variables Ri and Rj are summarized here :
- Correlation measures the strength of the linear relationship between two random variables.
- Correlation has no units.
- The correlation ranges from -1 to +1.
- If Corr(Ri,Rj) = 1, the random variables have perfect positive correlation. This means that a movement in one random variable results in a proportional positive movement in the other relative to its mean.
- If Corr(Ri,Rj) = -1, the random variables have perfect negative correlation. This means that a movement in one random variable results in an exact opposite proportional movement in the other relative to its mean.
- If Corr(Ri,Rj) = 0, there is no linear relationship between the variables, indicating that prediction of Ri cannot be made on the basis of Rj using linear methods.
L. Calculate and interpret the expected value, variance, and standard deviation of a random variable and of returns on a portfolio :
M. Calculate and interpret covariance given a joint probability function :
The expected value and variance for a portfolio of assets can be determined using the properties of the individual assets in the portfolio. To do this, it is necessary to establish the portfolio weight for each asset. As indicated in the formula, the weight, w, of portfolio asset i is simply the market value currently invested in the asset divided by the current market value of the entire portfolio :
Portfolio expected value. The expected value of a portfolio composed of n assets with weights, wi, and expected returns, Ri, can be determined using the following formula :
E(Rp) =w1 * E(R1) + w2 * E(R2) + ... + wn * E(Rn)
Portfolio variance. The variance of the portfolio return uses the portfolio weights also, but in a more complicated way :
N. Calculate and interpret an updated probability using Bayes’ formula :
Bayes’ formula is used to update a given set of prior probabilities for a given event in response to the arrival of new information. The rule for updating prior probability of an event is :
E(Rp) =w1 * E(R1) + w2 * E(R2) + ... + wn * E(Rn)
Portfolio variance. The variance of the portfolio return uses the portfolio weights also, but in a more complicated way :
N. Calculate and interpret an updated probability using Bayes’ formula :
Bayes’ formula is used to update a given set of prior probabilities for a given event in response to the arrival of new information. The rule for updating prior probability of an event is :
O. Identify the most appropriate method to solve a particular counting problem, and solve counting problems using factorial, combination, and permutation notations :
Labeling refers to the situation where there are n items that can each receive one of k different labels. The total number of ways that the labels can be assigned is :
where ! stands for factorial. For example : 4! = 4*3*2*1=24
The general expression for n factional is :
n! = n * (n -1) * (n – 2) * ... *1, where by definition 0! = 1
A special case of labeling arises when the number of labels equals 2 (k=2). In this case, we can let r=n1 and n2 = n – r since n = n1 + n2. The general formula for labeling when k=2 is called the combination formula and is expressed as :
Labeling refers to the situation where there are n items that can each receive one of k different labels. The total number of ways that the labels can be assigned is :
where ! stands for factorial. For example : 4! = 4*3*2*1=24
The general expression for n factional is :
n! = n * (n -1) * (n – 2) * ... *1, where by definition 0! = 1
A special case of labeling arises when the number of labels equals 2 (k=2). In this case, we can let r=n1 and n2 = n – r since n = n1 + n2. The general formula for labeling when k=2 is called the combination formula and is expressed as :
where nCr is the number of possible ways (combinations) of selecting r items from a set of n items when the order of selection is not important.
Another useful formula is the permutation formula. A permutation is a specific ordering of a group of objects. The question of how many different groups of size r in specific order can be chosen from n objects is answered by the permutation formula :
There are five guidelines that may be used to determine which counting method to employ when dealing with counting problems :
Another useful formula is the permutation formula. A permutation is a specific ordering of a group of objects. The question of how many different groups of size r in specific order can be chosen from n objects is answered by the permutation formula :
There are five guidelines that may be used to determine which counting method to employ when dealing with counting problems :
- The multiplication rule of counting is used when there are two or more groups. The key is that only one item may be selected from each group.
- Factorial is used by itself when there are no groups. Given n items, there are n! ways of arranging them.
- The labeling formula applies to three or more sub-groups of predetermined size. Each element of the entire group must be assigned a place, or label, in one of the three or more sub-groups.
- The combination formula applies to only two groups of predetermined size. Look for the word “choose” or “combination”.
- The permutation formula applies to only two groups of predetermined size. Look for a specific reference “order” being important.
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