2017, Study Session # 2, Reading # 9
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“PROBABILITY CONCEPTS”
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.
ΣP( 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(RIc
)
= 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.
2017, Study Session # 2, Reading # 9
Copyright © FinQuiz.com. All rights reserved.
Covariance
Measure of how two assets move together. 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).
Covariance
Variables tend to
+ ve⇒ Move in same direction.
- ve ⇒ Move in opposite
direction.
‘0’⇒ Asset returns are
unrelated.
Portfolio
=
()
Expected Value
=
+
Variance
⇒Where wi = market value of investment in asset ‘i’ market value of the portfolio
Conditional Expected Value
Calculated using conditionalprobabilities.
Are contingent upon the occurrence of some other event.
Expected Value
Probability weightedoutcomes of a random variable.
It is the best guess of the outcome of a random variable.
Value
Correlation
Variables tend to
+1 ⇒ Perfectly positive ⇒ Move proportionally in the same direction.
-1 ⇒ Perfectlynegative⇒ Move proportionally in the opposite direction.
0 ⇒ Uncorrelated⇒ No linear relationship.
Correlation
Measures the direction as well as the magnitude. It is a standardized measure of co-movement. 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.
2017, Study Session # 2, Reading # 9
Copyright © FinQuiz.com. All rights reserved.
Counting Methods
݊
!
݊
ଵ! …
݊
!
Labeling Formula
The number of ways ‘n’ objects can be labeled with k different labels.
Factorial [!]
Arranging a givenset of ‘n’ items.
No subgroups. There are n! ways
of arranging ‘n’ items.
Permutation [
nPr]
Number of ways of choosing r objects from a total of n objects when order matters.
Combination [
nC
r]
Choosing ‘r’ items from a set of ‘n’ items when order does not matter.
