1.3: probability, expected value, and variance

Question 1

why is the return on a risky asset is a random variable?

Answer 1

because the outcomes (possible values) are uncertain

Question 2

what is an event?

Answer 2

a specified set of outcomes (e.g., a return is less than 8%)

Question 3

The probability measures what?

Answer 3

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

Question 4

The sum of the probabilities of any set of mutually exclusive and exhaustive events is what?

what does this mean?

Answer 4

1

Question 5

what are mutually exclusive and exhaustive events in a set of probabilities ?

Answer 5

only one can occur

all possible outcomes are covered

Question 6

Three approaches are used to estimate probabilities

Answer 6

Subjective probabilities

Empirical probabilities

A priori probabilities

Question 7

Subjective probabilities

Answer 7

based on personal judgment

Question 8

Empirical probabilities

Answer 8

derived from relative frequencies from historical data

Question 9

when do Empirical probabilities work and when do they not?

Answer 9

Only works if relationships are stable through time

It will not be useful for very rare events

Question 10

A priori probabilities

Answer 10

deduced using logic rather than observation

Both empirical and a priori probabilities are considered objective because they are typically the same for all people

Question 11

odds

Answer 11

Odds for E = P(E) / (1 - P(E))

Question 12

The odds against an event happening

Answer 12

simply the reciprocal of the odds for that event

Question 13

Unconditional probabilities

Answer 13

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%

Question 14

Conditional probabilities

Answer 14

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%)

Question 15

A joint probability

Answer 15

reflects the probability of two or more outcomes occurring.

P(AB) represents the probability of both A and B occurring

Question 16

The conditional probability of A given B can be written mathematically as:

Answer 16

P(AB) = P(A|B) * (PB)

P(A|B) = P(AB)/P(B) for P(B) =/= 0

P(BA) = P(B|A) * P(A)

Question 17

The addition rule for probabilities formula

Answer 17

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)
.

Question 18

when are two events independent?

Answer 18

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)

Question 19

formula for independent events

Answer 19

P(AB) = P(A)*P(B)

Question 20

The complement of S

Answer 20

S^C

It represents event S not occurring

It follows P(S) + P(S^C)=1

Question 21

the total probability rule

Answer 21

P(A) = P(AS) + P(AS^C)

= P(A|S)P(S) + P(A|S^C)P(S^C)

Question 22

The expected value E(X)

Answer 22

the probability-weighted average of the possible outcomes of the random variable

Question 23

The expected value E(X) formula

Answer 23

E(X) = Sum of all [P(Xi)*Xi)

Xi: the outcome

P(Xi): probability of the outcome

Question 24

The variance, denoted (σ^2) of (X)

Answer 24

the probability-weighted average of the squared deviations from the random variable’s expected value

Question 25

A higher variance indicates what?

Answer 25

more dispersion in the random variable

Question 26

The standard deviation

Answer 26

the positive square root of the variance

Question 27

variance of a the weighted average of possible outcomes of on random variable

formula

Answer 27

σ^2(R) = Sum of all Wi[Xi - E(R)]^2

Wi: weight of return (probability of return)

Xi: return

E(R): Expected return

Question 28

Conditional expected values

Answer 28

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

Question 29

Conditional expected value formula

Answer 29

E(X|S) = Sum of all E(X|Sn) * P(Sn)

Question 30

the total probability rule for the expected value

Answer 30

The conditional expected values can be used to calculate the unconditional expected value

(X) = Sum of all [E(X|Sn)*P(Sn)

Question 31

formula for a conditional variance

Answer 31

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

Question 32

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.

Answer 32

A. mutually exclusive and collectively exhaustive.

Question 33

The addition rule for probabilities formula

Answer 33

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)