Book 2. Quantitative Analysis

FRM Part 1

QA 1. Fundamentals of Probability

Presented by: Sudhanshu

Module 1. Basics of Probability

Module 2. Conditional, Unconditional and Joint Probabilities

Module 1. Basics of Probability

Topic 1. Basics of Probability

Topic 2. Event and Event Spaces

Topic 3. Independent and Mutually Exclusive Events

Topic 4. Conditionally Independent Events

Topic 1.  Basics of Probability

• Random Variable: Unknown outcome (e.g., coin flip, tomorrow's temperature) described by probabilities of possible outcomes
• Probability Interpretation: Likelihood of occurrence; fair coin has P(heads) = 50% meaning 50 heads expected in 100 flips
• Probability Range: Must be between 0 (outcome won't happen) and 1 (outcome certain to happen)
• Conditional Probability P(A|B): Probability of A occurring given that B has already occurred (e.g., temperature 70-80°F given cloudy sky)
• Unconditional Probability: Marginal probability without conditions (e.g., temperature 70-80°F in Seattle on any day)
• Joint Probability P(AB): Probability that both events A and B will occur simultaneously

Topic 2. Event and Event Spaces

• Event: A single outcome or a combination of outcomes for a random variable.
  • Example: For a six-sided die roll:
    • Event: $x=3$, P(x=3) = 1/6.
    • Event: getting a 3 or 4, P(x=3 or x=4) = 2/6.
    • Event: getting an even number, P(x is even) = P(x=2, x=4, or x=6) = 3/6.
• Event Space: The set of all possible outcomes and combinations of outcomes for a random variable.
  • Example: For a fair coin flip, the event space includes head, tail, head and tail, and neither head nor tail.
    • P(head) = 50%, P(tail) = 50%.
    • P(both head and tail) = 0.
    • P(neither head nor tail) = 0.

Practice Questions: Q1

Q1. For the roll of a fair six-sided die, how many of the following are classified as events?

• The outcome is 3.
• The outcome is an even number.
• The outcome is not 2, 3, 4, 5, or 6.

A. One.

B. Two.

C. Three.

D. None.

Practice Questions: Q1 Answer

Explanation: C is correct.

All of the outcomes and combinations specified are included in the event space for the random variable.

Topic 3. Independent and Mutually Exclusive Events

• Definition: Two events where knowing the outcome of one does not affect the probability of the other.
  • Conditions for Independence:
    1.                                          (Joint probability is the product of unconditional probabilities).
    2.                               (Conditional probability of A given B is simply the unconditional probability of A).
  • Example: Flipping a coin twice. Getting heads on the first flip doesn't change the probability of getting heads on the second.
• Mutually Exclusive Events: Two events that cannot both happen simultaneously.
  • Example: For a single die roll, "x = an even number" and "x = 3" are mutually exclusive.
  • If A and B are mutually exclusive, $P\left(AB\right)=0$.
• General Addition Rule: $P\left(A\text{ or}B\right)=P\left(A\right)+P\left(B\right)-P\left(AB\right)$.
  • ​For mutually exclusive events: P(A or B)=P(A)+P(B)

Practice Questions: Q2

Q2. Which of the following equalities does not imply that the events A and B are independent?

A. P(AB) = P(A) × P(B).

B. P(A or B) = P(A) + P(B) – P(AB).

C. P(A|B) = P(A).

D. P(AB) / P(B) = P(A).

Practice Questions: Q2 Answer

Explanation: B is correct.

P(A or B) = P(A) + P(B) – P(AB) holds for both independent and dependent events. The other equalities are only true for independent events.

Topic 4. Conditionally Independent events

• Definition: Two conditional probabilities, $P(A|C)$ and $P(B|C)$, are conditionally independent if
• Key Distinction: Events can be independent (unconditionally) but conditionally dependent, or vice versa.
• Example:
  • Event A: "scores above average on an exam."
  • Event B: "is taller than average."
  • For grade school students, P(A) and P(B) might be dependent (taller students tend to be older and score better).
  • However, if we introduce Event C: "age equals 8," then $P(A|C)$ and $P(B|C)$ might be independent (among 8-year-olds, height might not correlate with exam scores).

Practice Questions: Q3

Q3. Two independent events:

A. must be conditionally independent.

B. cannot be conditionally independent.

C. may be conditionally independent or not conditionally independent.

D. are conditionally independent only if they are mutually exclusive events.

Practice Questions: Q3 Answer

Explanation: C is correct.

Two independent events may be conditionally independent or not conditionally independent.  

Module 2. Conditional, Unconditional, And Joint Probabilities

Topic 1. Discrete Probability Function

Topic 2. Conditional and Unconditional Probabilities

Topic 3. Bayes’ Rules

Topic 1.  Discrete Probability Function

• Definition: A function for which there are a finite number of possible outcomes, and it provides the probability of each outcome.
• Properties:
  • The probability of each outcome must be between 0 and 1.
  • The sum of the probabilities of all possible outcomes must equal 1 (or 100%).
• Example: Random variable with outcomes x = 1, 2, 3, or 4, and $P\left(x\right)=x/10$.
  • $P(x=1)=1/10$
  • $P(x=2)=2/10$
  • $P(x=3)=3/10$
  • $P(x=4)=4/10$
  • $P(x=2\text{ or}4)=P(x=2)+P(x=4)=2/10+4/10=6/10=60\%$.
  • Sum of probabilities = $1/10+2/10+3/10+4/10=10/10=100\%$

Practice Questions: Q4

Q4. The probability function for the outcome of one roll of a six-sided die is given as P(X)= x/21. What is P(x > 4)?

A. 16.6%.

B. 23.8%.

C. 33.3%.

D. 52.4%.

Practice Questions: Q4 Answer

Explanation: D is correct.

The probability of x > 4 is the probability of an outcome of 5 or 6 (5/21 + 6/21 =52.4%).

Topic 2.  Conditional and Unconditional Probabilities

• Unconditional Probability (Marginal Probability): The probability of an event occurring without any other information.
  • Example: The probability that a day's high temperature in Seattle will be between 70 and 80 degrees.
• Conditional Probability: The probability of an event occurring given that some other event has already occurred.
  • Notation: $P(A|B)$ (Probability of A given B).
  • Example: The probability that the high temperature will be between 70 and 80 degrees, given that the sky is cloudy that day.
• Joint Probability: The probability that both Event A and Event B will occur, denoted as $P(AB)$.
  • Relationship: 
  • Rearrangement: 

​

• Total Probability Rule: If A and Bi​ are mutually exclusive and exhaustive events, then

Practice Questions: Q5

Q5. The relationship between the probability that both Event A and Event B will occur and the conditional probability of Event A given that Event B occurs is:

A.

B.

C.

D.  

Practice Questions: Q5 Answer

Explanation: A is correct.

The (joint) probability that both A and B will occur is equal to the conditional probability of Event A given that Event B has occurred, multiplied by the unconditional probability of Event B.  

Topic 3.  Bayes' Rule

• Purpose: Allows us to update our estimates of the unconditional probability of an event using new information (the outcome of another event).
• Formula:

​ 

• Where $P(B)$ can be expanded using the total probability rule:
                                                                         (where Ac is the complement of A)
• Intuition: It links the conditional probability of A given B to the conditional probability of B given A, incorporating the prior (unconditional) probabilities of A and B.
• Application Example (Stock Gains & Economy):
  • Given: $P\left(\text{Outperform}\right)=60\%$, $P(\text{Gains}|\text{Outperform})=70\%$.
  • Given: $P\left(\text{Underperform}\right)=40\%$, $P(\text{Gains}|\text{Underperform})=20\%$.
  • Calculate $P(\text{Outperform}|\text{Gains})$.
  • $P\left(\text{Outperform and Gains}\right)=0.7\times 0.6=0.42$ 
  • $P\left(\text{Underperform and Gains}\right)=0.2\times 0.4=0.08$ 
  • $P\left(\text{Gains}\right)=P\left(\text{Outperform and Gains}\right)+P\left(\text{Underperform and Gains}\right)=0.42+0.08=0.50$ 
  •  

Practice Questions: Q6

Q6. The probability that shares of Acme will increase in value over the next month is 50% and the probability that shares of Acme and shares of Best will both increase in value over the next month is 40%. The probability that Best shares will increase in value, given that Acme shares increase in value over the next month, is closest to:

A. 20%.

B. 40%.

C. 80%.

D. 90%.

Practice Questions: Q6 Answer

Explanation: C is correct.

Bayes’ formula tells us that:

Applying that to the information given, we can write: