1.3: probability, expected value, and variance Flashcards
why is the return on a risky asset is a random variable?
because the outcomes (possible values) are uncertain
what is an event?
a specified set of outcomes (e.g., a return is less than 8%)
The probability measures what?
the chance a specified event will occur
The probability of any event is between 0 and 1
If the event is impossible, the probability is 0. If an event is certain, the probability is 1
The sum of the probabilities of any set of mutually exclusive and exhaustive events is what?
what does this mean?
1
what are mutually exclusive and exhaustive events in a set of probabilities ?
only one can occur
all possible outcomes are covered
Three approaches are used to estimate probabilities
Subjective probabilities
Empirical probabilities
A priori probabilities
Subjective probabilities
based on personal judgment
Empirical probabilities
derived from relative frequencies from historical data
when do Empirical probabilities work and when do they not?
Only works if relationships are stable through time
It will not be useful for very rare events
A priori probabilities
deduced using logic rather than observation
Both empirical and a priori probabilities are considered objective because they are typically the same for all people
odds
Odds for E = P(E) / (1 - P(E))
The odds against an event happening
simply the reciprocal of the odds for that event
Unconditional probabilities
based on the universe of all possibilities
They can be thought of as stand-alone probabilities.
–> For example, an investor could calculate the probability that the return is less than 8%
Conditional probabilities
restrict the set of possibilities
–> For example, an investor could calculate the probability that the return is less than 8% given it is positive
This is how investors incorporate new information (return is at least 0%)
A joint probability
reflects the probability of two or more outcomes occurring.
P(AB) represents the probability of both A and B occurring
The conditional probability of A given B can be written mathematically as:
P(AB) = P(A|B) * (PB)
P(A|B) = P(AB)/P(B) for P(B) =/= 0
P(BA) = P(B|A) * P(A)
The addition rule for probabilities formula
P(A or B) = P(A) + P(B) − P(AB)
Note that A∩B is the event of both A and B occurring, so the probability is represented by P(AB)
.
when are two events independent?
if the outcome of one does not affect the other
Mathematically, events A and B are independent if and only if
P(A∣B) = P(A)
or,
P(B∣A)=P(B)
formula for independent events
P(AB) = P(A)*P(B)
The complement of S
S^C
It represents event S not occurring
It follows P(S) + P(S^C)=1
the total probability rule
P(A) = P(AS) + P(AS^C)
= P(A|S)P(S) + P(A|S^C)P(S^C)
The expected value E(X)
the probability-weighted average of the possible outcomes of the random variable
The expected value E(X) formula
E(X) = Sum of all [P(Xi)*Xi)
Xi: the outcome
P(Xi): probability of the outcome
The variance, denoted (σ^2) of (X)
the probability-weighted average of the squared deviations from the random variable’s expected value
A higher variance indicates what?
more dispersion in the random variable
The standard deviation
the positive square root of the variance
variance of a the weighted average of possible outcomes of on random variable
formula
σ^2(R) = Sum of all Wi[Xi - E(R)]^2
Wi: weight of return (probability of return)
Xi: return
E(R): Expected return
Conditional expected values
take into account new information or events.
The conditional expected value of the random variable X given an event or scenario S is denoted E(X|S)
The conditional expected value of the random variable X is the probability-weighted average of the possible outcomes of the random variable conditional on S
Conditional expected value formula
E(X|S) = Sum of all E(X|Sn) * P(Sn)
the total probability rule for the expected value
The conditional expected values can be used to calculate the unconditional expected value
(X) = Sum of all [E(X|Sn)*P(Sn)
formula for a conditional variance
σ^2(X|S) = Sum of all Wi[Xi - E(Xi|S)]^2
Wi: weight of return (probability of return)
Xi: return
E(Xi|S): Expected conditional return
You roll a fair, 6-sided die. The potential outcomes are {1, 2, 3, 4, 5, 6}, each with probability 1/6. These 6 outcomes are most accurately described as:
A. mutually exclusive and collectively exhaustive.
B. collectively exhaustive, but not mutually exclusive.
C. mutually exclusive, but not collectively exhaustive.
A. mutually exclusive and collectively exhaustive.
The addition rule for probabilities formula
P(A or B) = P(A) + P(B) − P(AB)
The last term P(AB) must be subtracted to avoid double-counting
–> A∩B = P(AB)
