QTA 4 - MULTIVARIATE RANDOM VARIABLES Flashcards
How can a probability matrix be used in relation to a probability mass function (PMF)?
A probability matrix relates realizations to probabilities and serves as a tabular representation of a PMF
It describes discrete distributions defined over a finite set of values.
What does the PMF of a bivariate random variable represent?
The PMF returns the probability that two random variables each take a certain value
It requires three axes: X1, X2, and the probability mass/density.
Define covariance.
Covariance is a measure of how two random variables move together.
What does the expectation of a function for a bivariate discrete random variable represent?
It is a probability weighted average of the function of the outcomes.
How is the marginal PMF of a bivariate random variable computed?
It is computed by summing the joint PMF across all values of the other variable.
What is a marginal distribution?
The distribution of a single component of a bivariate random variable.
What condition must be met for two random variables to be independent?
The joint PMF must equal the product of the marginal PMFs.
What is the relationship between covariance and correlation?
Covariance measures the direction of the relationship, while correlation measures the strength and direction of the relationship.
What is the formula for the PMF of a trinomial random variable?
fX1,X2 = (n! / (x1! x2! (n - x1 - x2)!)) * p1^x1 * p2^x2 * (1 - p1 - p2)^(n - x1 - x2)
What is the definition of a conditional distribution?
It summarizes the probability of outcomes for one random variable given that another takes a specific value.
Fill in the blank: The expectation of a function g(X1, X2) is defined as E[g(X1, X2)] = ___
ΣΣ g(x1, x2)fX1,X2
What does the CDF of a bivariate variable return?
It returns the total probability that each component is less than or equal to a given value.
True or False: The components of a bivariate random variable are independent if the joint PMF is equal to the sum of the marginal PMFs.
False
What is the significance of the i.i.d property in random variables?
It is helpful in computing the mean and variance of a sum of i.i.d random variables.
How is the variance of a weighted sum of two random variables computed?
It involves the variances and covariances of the random variables.
What are the two components of a bivariate random variable?
X1 and X2.
What is the relationship between the marginal PMF and the marginal CDF?
The marginal CDF is defined using the marginal PMF to measure the total probability less than a given value.
What does the first moment of X represent?
The mean E[X] = [μ1, μ2].
What is the definition of the conditional probability of two events?
P(A|B) = P(A ∩ B) / P(B).
True or False: Knowledge about the value of X2 must contain information about X1 for independence.
False
What is the formula for covariance between X1 and X2?
Cov[X1, X2] = E[(X1 - E[X1])(X2 - E[X2])] = E[X1X2] - E[X1]E[X2].
How is the conditional PMF computed for a bivariate random variable?
It is the joint probability divided by the marginal probability of the conditioning variable.
What is the definition of covariance?
Covariance is a measure of dispersion that captures how the variables move together.
How is the covariance between two variables (X_1) and (X_2) defined?
Cov[X1, X2] = E[(X1 - E[X1])(X2 - E[X2])] = E[X1X2] - E[X1]E[X2]
What does the covariance of a variable with itself represent?
The covariance of a variable with itself is just the variance of that variable.
In a bivariate random variable, how many variances and covariances are there?
There are two variances and one covariance.
What is the common abbreviation for the variance of (X_1) and (X_2)?
V[X1] is denoted as σ² and V[X2] as σ².
What does the symbol σ12 represent?
The covariance between (X_1) and (X_2).
What is the range of values that covariance can take?
Covariance can take on values from -∞ to +∞.
Why is correlation often reported instead of covariance?
Correlation is a scale-free measure and is easier to interpret.
How is correlation between two variables obtained?
By dividing their covariance by their respective standard deviations.
What does correlation measure?
The strength of the linear relationship between two variables.
What is the range of correlation values?
Correlation is always between -1 and 1.
What does a positive correlation indicate?
When (X_1) and (X_2) tend to increase together.
What does a negative correlation indicate?
If (X_2) tends to decrease when (X_1) increases.
How do location shifts affect covariance?
Location shifts have no effect on covariance.
How does scaling affect covariance?
The scale of each component contributes multiplicatively to the change in covariance.
What is the formula for covariance after applying shifts and rescaling?
Cov[a + bX1, c + dX2] = bd Cov[X1, X2]
What is the relationship between correlation and covariance when scaling?
Corr[aX1, bX2] = sign(a) sign(b) Corr[X1, X2]
When is the covariance between two independent random variables zero?
Cov[X1, X2] = 0.
What is the relationship between covariance and variance of a random variable?
Cov[X1, X1] = Var(X1).
What are the variance formulas when (X_1) and (X_2) are not independent?
V[X1 + X2] = V[X1] + V[X2] + 2 Cov[X1, X2] and V[X1 - X2] = V[X1] + V[X2] - 2 Cov[X1, X2].
What does the correlation coefficient (ρ) indicate?
The strength and direction of a linear relationship between two random variables.
What does a correlation of ρ = -1 indicate?
A perfect negative relationship.
What does a correlation of ρ = 0 indicate?
No linear relationship.
What does a correlation of ρ = 1 indicate?
A perfect positive relationship.
What happens to correlation when two random variables are independent?
They must have zero correlation.
Can two variables have zero correlation and still be dependent?
Yes, they can be dependent without having a linear relationship.
What is coskewness?
The third cross central moment indicating simultaneous extreme deviations of two random variables.
What is cokurtosis?
The fourth cross central moment indicating simultaneous extreme positive and negative deviations of two random variables.
What is a conditional expectation?
An expectation when one random variable takes a specific value or falls into a defined range.
What is a conditional variance?
The variance of a random variable given another variable.
What can cause dependence between random variables in finance?
Shifts in investor risk aversion, cross-asset spillovers, crowded portfolio strategies.
What is the role of conditioning in random variables?
Conditioning helps to remove the dependence between variables.
How does the transition from discrete to continuous random variables affect calculations?
It involves replacing PMF with PDF and sums with integrals.
What is the form of a probability density function (PDF) for continuous random variables?
The PDF has the same form as a PMF and is written as 𝑓X1,X2(𝑥1, 𝑥2)
How is the probability in a rectangular region defined for continuous random variables?
Pr(𝑙1 < 𝑋1 < 𝑢1 ∩ 𝑙2 < 𝑋2 < 𝑢2) = ƒ ƒ 𝑓X1,X2(𝑥1, 𝑥2) 𝑑𝑥1𝑑𝑥2
What does the cumulative distribution function (CDF) represent in the context of continuous random variables?
The CDF is the area of a rectangular region under the PDF where the lower bounds are —∞ and the upper bounds are the arguments in the CDF.
What must a PDF function always do?
A PDF function is always non-negative and must integrate to one across the support of the two components.
What is the simplest form of continuous multivariate random variable?
The standard bivariate uniform where the two components are independent.
What is the PDF of the standard bivariate independent uniform random variable?
𝑓X1,X2 = 𝑘, where the support of the density is the unit square.
What is the CDF of the standard bivariate independent uniform random variable?
𝐹X1,X2 = 𝑥1𝑥2
True or False: The transition from discrete multivariate random variables to continuous ones primarily changes the PMF to a PDF.
True
Fill in the blank: The most substantial change when moving from discrete to continuous random variables is the switch from sums to _______.
integrals
What must be adjusted if the PDF is not defined for all values?
The lower bounds can be adjusted to be the smallest values where the random variable has support.
What denotes conditional independence in the context of distributions?
This distribution is therefore conditionally independent even though the original distribution is not.
