Probability Concepts Flashcards
Define a random variable, an outcome, and an event
A random variable is an uncertain quantity/number.
e.g the number you get when rolling a dice
An outcome is an observed value of a random variable.
e.g rolling a 4
An event is a single outcome or a set of outcomes.
e.g roling a 4
Mutually exclusive events are events that cannot both happen at the same time.
1-6 are mutually exclusive (cant roll 2 numbers)
Exhaustive events are those that include all possible outcomes.
1-6 is exhaustive (cant roll a 7)
The probability of occurrence of any event (Ei) is between 0 and 1 (i.e., 0 ≤ P(Ei) ≤ 1).
e.g it must sum to 1
If a set of events, E1, E2, … En, is mutually exclusive (cant both happen at same time) and exhaustive (covers all events), the probabilities of those events sum to 1 (i.e., ΣP(Ei) = 1).
e.g the probability of the events must sum to 1
Define the different types of probability
An empirical probability is established by analyzing past data (outcomes).
An a priori probability is determined using a formal reasoning and inspection process (not data). Inspecting a coin and reasoning that the probability of each side coming up when the coin is flipped is an example of an a priori probability.
A subjective probability is the least formal method of developing probabilities and involves the use of personal judgment. An analyst may know many things about a firm’s performance and have expectations about the overall market that are all used to arrive at a subjective probability, such as “I believe there is a 70% probability that Acme Foods will outperform the market this year.”
Empirical and a priori probabilities, by contrast, are objective probabilities.
Explain the Joint, addition and multiplication rules of probability
And probability weighted outcome / average
Multiplication Rule: P(AB) = P(A | B) × P(B)
Intuition: if when B occurs, A occurs 50% of time. Multiply them
Addition rule: P(A or B) = P(A) + P(B) − P(AB) (do not double count)
So PA + PB - (PA x PB)
Total probability rule: When all outcomes are mutually exclusive and exhaustive, add them all up to get back to the unconditional probability.
Multiply the unconditional probability by the outcome e.g prob of 24% x outcome of receiving 50k = 12k
Explain the total probability rule
The total probability rule is used to determine the unconditional probability of an event, given conditional probabilities.
total probability rule: P(R) = P(R | S1) × P(S1) + P(R | S2) × P(S2) + . . . + P(R | SN) × P(SN)
Calculate and explain expected value
Probability weighted sum of all possible outcomes
Calculate portfolio expected return
In simple terms, the portfolio expected return is a way to estimate the average return you can expect from your investment portfolio. It takes into account the expected returns of individual stocks or assets within the portfolio and the proportions in which you have allocated your money to them.
Step 1: Calculate the weighted return of each stock ( Weight (decimal) x expected return %
Step 2: Sum the expected return for each stock
Calculate covariance
Step 1: Calculate the means of X and Y.
Add all the values and divide by the number of observations to find the mean
Step 2: For each observation, calculate it’s deviation from the mean. u - X
Step 3: Multiply the deviations. EG for Observation 1, multiply the deviations of X x Y
Step 4: Sum the deviations.
Step 5: Divide by n-1
Example:
Suppose we have the following set of paired observations for variables X and Y:
X: 2, 4, 6, 8, 10
Y: 3, 5, 7, 9, 11
Step 1: Gather the data - We have the paired observations for X and Y.
Step 2: Calculate the means:
μₓ = (2 + 4 + 6 + 8 + 10) / 5 = 6
μᵧ = (3 + 5 + 7 + 9 + 11) / 5 = 7
Step 3: Calculate the deviations:
(Xᵢ - μₓ) and (Yᵢ - μᵧ)
(2 - 6) = -4 (3 - 7) = -4
(4 - 6) = -2 (5 - 7) = -2
(6 - 6) = 0 (7 - 7) = 0
(8 - 6) = 2 (9 - 7) = 2
(10 - 6) = 4 (11 - 7) = 4
Step 4: Multiply the deviations:
(-4)(-4) = 16
(-2)(-2) = 4
(0)(0) = 0
(2)(2) = 4
(4)(4) = 16
Step 5: Sum the products:
16 + 4 + 0 + 4 + 16 = 40
Step 6: Divide by (n-1):
40 / (5 - 1) = 40 / 4 = 10
Therefore, the covariance between X and Y for this example is 10.
Calculate variance. What is the difference between calculating variance and sample variance?
Data: 2, 4, 6, 8, 10
Step 1: Calculate the mean
Compute the mean (average) of the dataset. Add up all the values and divide by the total number of data points:
Mean (μ) = (2 + 4 + 6 + 8 + 10) / 5 = 6
Step 2: Calculate the deviations
Calculate the deviation of each data point from the mean. Subtract the mean from each data point:
Deviation = Data - Mean
Deviation from 2: 2 - 6 = -4
Deviation from 4: 4 - 6 = -2
Deviation from 6: 6 - 6 = 0
Deviation from 8: 8 - 6 = 2
Deviation from 10: 10 - 6 = 4
Step 4: Square the deviations
Square each deviation obtained in step 3:
Squared Deviation = Deviation^2
Squared Deviation from -4: (-4)^2 = 16
Squared Deviation from -2: (-2)^2 = 4
Squared Deviation from 0: (0)^2 = 0
Squared Deviation from 2: (2)^2 = 4
Squared Deviation from 4: (4)^2 = 16
Step 5: Calculate the variance
Compute the variance by taking the average of the squared deviations. Sum up all the squared deviations and divide by the total number of data points:
Variance = Σ(Squared Deviation) / n
Variance = (16 + 4 + 0 + 4 + 16) / 5 = 40 / 5 = 8
Therefore, the variance of the dataset {2, 4, 6, 8, 10} is 8.
Calculate Covariance using Correlation
Calculate correlation using covariance and standard deviation
Covariance = Correlation x (SDX x SDY)
Correlation = CovAB / SD 1 x SD 2
Calculate portfolio variance
With Covariance
(Variance A x Weight A) + (Variance B x Weight B) + (2 x Weight A x Weight B) x (Covariance AB)
With Correlation
(Variance A x Weight A) + (Variance B x Weight B) + (2 x Weight A x Weight B) x (Correlation AB x Variance A x Variance B)
How do you interpret a covariance matrix
- Calculate the expected returns for A and B
- Multiply the probability by the Return’s difference from the mean for A and B
- Multiply A and B for each return
Labelling / Factorial
Consider a portfolio consisting of eight stocks. Your goal is to designate four of the stocks as “long-term holds,” three of the stocks as “short-term holds,” and one stock as “sell.” How many ways can these eight stocks be labeled?
8!
/
4! x 3! x 1!
Labelling / Factorial
When would you use the combination formula?
When there are 2 labelling options & order is not important
How many ways can three stocks be sold from an 8-stock portfolio?
8 (2nd) nCr 3 = 56
Labelling / Factorial
When would you use the permutation formula?
When there are 2 labelling options & order is important
How many ways can three stocks be sold from an 8-stock portfolio?
8 (2nd) nPr 3 = 336
