2) Multivariate Random Variables Flashcards
When are two events independent
P(A ∩ B) = P(A)P(B)
What is the joint cumulative distribution function
When are two bivariate random variable independent
What is a discrete bivariate random variable
If the range of a bivariate random variable (X, Y ) can only assume countably many values, RX,Y = {(xi, yi) | i ⩾ 1}
What is the joint probability mass function
What are the properties of the joint pmf
What are the marginal probability mass functions for discrete bivariate random variables
What is the joint probability density function
fx,y such that
What are the properties of the joint pdf
What are the marginal density functions for continuous bivariate random variables
If (X, Y ) ∼ N(µX, µY ; σ2X, σ2Y; ρ), what do we know about the marginal distributions of X and Y
The marginal distributions of X and Y are given by X ∼ N(µX, σ2X) and Y ∼ N(µY , σ2Y)
If (X, Y ) ∼ N(µX, µY ; σ2X, σ2Y; ρ), when are X and Y indepedent
If and only if ρ = 0
Suppose that X and Y are independent discrete random variables with probability mass functions pX(x) and pY (y), respectively. Then, what is the pmf of Z = X + Y
What is the Convolution theorem
What does it mean for a random variable to be independent
and identically distributed
The random variables are independent and all have the same distribution (so that FX1 = FX2 = · · · = FXn)
What is the the conditional probability of A given B
What is the Law of Total Probability
What is the conditional probability mass function
What is the conditional distribution function for discrete variables
What is the conditional distribution function for continuous variables
What is the conditional joint probability density function for continuous random variables
When are bivariate discrete random variables independent in terms of their conditional pmf
When are bivariate continuous random variables independent in terms of their conditional pdf
How do you find the joint density fU,V (u,v) of a transformed bivariate continuous random variable (U,V)
