Prob B

Question 1

What is a random variable?

Answer 1

A quantity that is measured in an experiment with a random outcome, whose value depends of the experiment

Question 2

Define a discrete distribution for a random variable

Answer 2

Question 3

Define a continuous distribution for a random variable

Answer 3

Question 4

Define the cumulative distribution function for both discrete and continuous functions

Answer 4

Question 5

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

Answer 5

Question 6

Define a joint distribution of X and Y

Answer 6

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)

Question 7

Define the probability of a joint distribution for discrete random variables

Answer 7

Question 8

Define the probability of a joint distribution for continuous random variables

Answer 8

Question 9

When are two random variables independent, state in terms of cumulative distributions

Answer 9

Question 10

Define a convolution for mass and density functions

Answer 10

Question 11

Define the exponential variable with parameter alpha

Answer 11

Question 12

State and prove the memoryless property for an exponential variable

Answer 12

Question 13

What is the gamma distribution with parameters n,alpha, and give the notation

Answer 13

Question 14

Answer 14

Question 15

Define the poisson counting process

Answer 15

Question 16

Show by looking at P(N_t >= k) that N_t is a poisson random variable

Answer 16

Question 17

Define the expectation for a discrete distribution and a continuous distribution

Answer 17

Question 18

Define the expectation of a discrete distribution for g(X) with f as mass function and g some given function. Prove

Answer 18

Question 19

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)

Answer 19

Question 20

Define the bernoulli random variable

Answer 20

Question 21

What is a moment generating function

Answer 21

Question 22

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

Answer 22

Question 23

Prove that expectations are linear

Answer 23

Question 24

State Fubini’s Theorem

Answer 24

If X and Y are independent then E(g(x)h(y)) = E(g(x))E(h(y))

Question 25

State and Prove the Cauchy-Schwartz Inequality

Answer 25

Question 26

Give the equation for the variance

Answer 26

Question 27

State the Covariance and prove it equals E(XY) - E(X)E(Y)

Answer 27

Question 28

Define Correlation

Answer 28

Question 29

Give the pdf of a gaussian distribution, and state the mean and variance

Answer 29

Question 30

Whats the MGF of a gaussian distribution

Answer 30

Question 31

Give the MGF, expectation and variance of a gamma distribution

Answer 31

Question 32

Give the MGF, expectation and variance of a poisson distribution with parameter lamda

Answer 32

Question 33

Give the MGF, expectation and variance of a binomial distribution with parameters n and p

Answer 33

Question 34

Give the MGF, expectation and variance of a uniform distribution between a and b

Answer 34