3) Expectation

Question 1

What is the expectation of a discrete random variable

Answer 1

Question 2

What is the expectation of a continuous random variable

Answer 2

Question 3

What is the variance of a random variable X

Answer 3

Question 4

What is the expectation of a function g(X) for discrete and continuous

Answer 4

Question 5

What is the expectation of a function g(X,Y) for discrete and continuous

Answer 5

Question 6

How can the expectation E[g(X,Y)] be expressed when g is just a function of x

Answer 6

Question 7

What is the linearity of expectation for two functions 𝑔1 and g2 and real numbers α and β

Answer 7

Question 8

What is the expectation of the product of two functions g(X) and h(Y) for independent random variables X and Y

Answer 8

Question 9

What is the variance of the linear combination αX+βY for independent random variables X and Y

Answer 9

Question 10

What is the covariance between random variables X and Y

Answer 10

Question 11

What are the properties of covariance

Answer 11

Question 12

What is the variance of the linear combination αX+βY for dependent random variables X and Y

Answer 12

Question 13

What is the variance of the linear combination ∑aiXi for dependent random variables X1 ,…,X and constants a1,…,an

Answer 13

Question 14

What is the variance of the linear combination ∑aiXi for independent random variables X1 ,…,X and constants a1,…,an

Answer 14

Question 15

What is the Cauchy-Schwarz inequality

Answer 15

Question 16

What is the correlation coefficient

Answer 16

Question 17

When are two random variables uncorrelated

Answer 17

If Cov(X, Y ) = 0
(Independent random variables are uncorrelated)

Question 18

What is the the conditional expectation of X given Y = y, for discrete random variable (X, Y )

Answer 18

Question 19

What is the the conditional expectation of X given Y = y, for continuous random variable (X, Y )

Answer 19

Question 20

What is the the conditional expectation of g(X, Y ) given Y = y

Answer 20

Question 21

What is the Law of total expectation

Answer 21

Question 22

How is the law of total expectation expressed for continuous and discrete random variables

Answer 22

Question 23

What is the conditional expectation of[α(X)g1(X, Y ) + β(X)g2(X, Y ) given X

Answer 23

Question 24

What is the conditional expectation E[g(X)∣Y] when X and Y are independent

Answer 24

Question 25

What is the conditional variance of Y given X = x

Answer 25

Question 26

What is the law of total variance

Answer 26