Probability in terms of Odds for the event Odds against the event Probability of non-occurrence divided by probability of occurrence.. Probability of occurrence divided by probabil
Trang 1“PROBABILITY CONCEPTS”
Random Variable
Quantity with
uncertain possible
value(s)
Outcome
An observed value
of a random variable
Event
A single outcome
or a set of outcomes
Mutually Exclusive Events
Both cannot happen at the same time
P(A|B) = 0 &
P(AB) = P(A|B) × P(B) = 0
Exhaustive Events
Include all possible outcomes
Two Defining Properties of
Probability
0 ≤ P(E) ≤ 1
i.e., Probability of an
event lies b/w 0 & 1
Σ P( E i ) = 1
i.e., Total probability is equal
to 1
Probability in terms of
Odds for the
event
Odds against the event
Probability of non-occurrence divided by probability of occurrence
Probability of
occurrence divided by
probability of
non-occurrence
Probability
Empirical Probability
Based on historical facts
or data
No judgments involved
Historical + non random
A Priori Probability
Based on logical analysis.
Random + historical.
Subjective Probability
An informal guess
Involves personal judgment
Objective Probability
Total Probability Rule
It highlights the relationship b/w unconditional & conditional
probabilities of mutually exclusive & exhaustive events
P(R) = P(RI) + P(RI c
)
= P(R|I) × P(I) + P(R|Ic) × P(Ic)
Addition Rule
Probability that at least one event will occur
P(A or B) = P(A) + P(B) - P(AB)
⇒ For mutually exclusive events
P(A or B) = P(A) + P(B)
Multiplication Rule
(Joint Probability)
Probability that both events will
occur
P(AB) = P(A|B) × P(B)
⇒ For mutually exclusive events;
P(A|B) = 0, hence,
P(AB) = 0
Unconditional Probability
Marginal probability.
Probability of occurrence of an event-regardless of the past or future occurrence.
Conditional Probability; P(A|B)
Probability of the occurrence of an event is affected by the occurrence of another event.
It is also known as likelihood of an occurrence.
‘|’ denotes ‘given’ or ‘conditional’ upon.
P(A|B) = P (AB)
P(B)
Mutually exclusive events P(A|B) = 0.
For independent events, P(A|B) = P(A)
Independent Events
Events for which occurrence of one has no effect on occurrence of the other.
P(A|B) = P(A)
P(B|A) = P(B)
Trang 2Covariance
It measures only direction
-∝≤ Cov(x, y) ≤ +∝(property)
It is measured in squared units
Cov(Ri,Rj) = E {[Ri - E(Ri)] [Rj – E(Rj)]}
= Σ P(S) [Ri – E(Ri)] [Rj – E(Rj)
Cov (RA,RA ) = variance (RA) (property)
direction
unrelated
Portfolio
()
Expected Value
Variance
⇒Where wi = market value of investment in asset ‘i’
market value of the portfolio
Conditional Expected Value
Calculated using conditional probabilities
occurrence of some other event
Expected Value
outcomes of a random variable
It is the best guess of the outcome of a random variable
Value Correlation Variables tend to
same direction
opposite direction
Correlation
It has no units
-1 ≤ corr (Ri,Rj) ≤ + 1
Corr (Ri,Rj) = Cov (Ri,Rj)
σ (Ri) σ (Rj)
Baye’s Formula
⇒Used to update a given set of prior probabilities in response to the arrival of new information
Updated probability prior Probability = of new info × probability of the unconditional event
probability of
new info
Trang 3Counting Methods
݊ !
݊ଵ! … ݊!
Labeling Formula
The number of ways ‘n’
objects can be labeled
with k different labels
Factorial [!]
Arranging a given set of ‘n’ items
of arranging ‘n’
items
Permutation [nPr] Number of ways of choosing r objects from
a total of n objects when order matters
Combination [nCr]
Choosing ‘r’ items from a set of ‘n’
items when order does not matter
Multiplication Rule
The number of ways k tasks can be done is (n1)(n2)(n3)…(ni)
... directionopposite direction
Correlation
It has no units
-1 ≤ corr (Ri,Rj) ≤ +
Corr (Ri,Rj) = Cov (Ri,Rj)...
Multiplication Rule
The number of ways k tasks can be done is (n1< /small>)(n2)(n3)…(ni)