Prob B Flashcards
What is a random variable?
A quantity that is measured in an experiment with a random outcome, whose value depends of the experiment
Define a discrete distribution for a random variable
Define a continuous distribution for a random variable
Define the cumulative distribution function for both discrete and continuous functions
If Y = g(X), where X is a random variable and g: R to R, then what is the cumulative distribution of y with respect to Y, and the probability density function relationships
Define a joint distribution of X and Y
The join distribution of two random variables is defined on a sample space Omega with a probability measure P mapping B to P((X,Y) in B)
Define the probability of a joint distribution for discrete random variables
Define the probability of a joint distribution for continuous random variables
When are two random variables independent, state in terms of cumulative distributions
Define a convolution for mass and density functions
Define the exponential variable with parameter alpha
State and prove the memoryless property for an exponential variable
What is the gamma distribution with parameters n,alpha, and give the notation
Define the poisson counting process
Show by looking at P(Nt >= k) that Nt is a poisson random variable
Define the expectation for a discrete distribution and a continuous distribution
Define the expectation of a discrete distribution for g(X) with f as mass function and g some given function. Prove
If the mass function is f for a continuous random random variable and g is a given function what is the expectation of g(X)
Define the bernoulli random variable
What is a moment generating function
If X and Y are random variables on the same sample space and g: R2 to R2 what is the expectation of g(X,Y) for both discrete and continuous cases
Prove that expectations are linear
State Fubini’s Theorem
If X and Y are independent then E(g(x)h(y)) = E(g(x))E(h(y))
State and Prove the Cauchy-Schwartz Inequality
Give the equation for the variance
State the Covariance and prove it equals E(XY) - E(X)E(Y)
Define Correlation
Give the pdf of a gaussian distribution, and state the mean and variance
Whats the MGF of a gaussian distribution
Give the MGF, expectation and variance of a gamma distribution
Give the MGF, expectation and variance of a poisson distribution with parameter lamda
Give the MGF, expectation and variance of a binomial distribution with parameters n and p
Give the MGF, expectation and variance of a uniform distribution between a and b
