Discrete & Continuous Distributions

Question 1

General formula for E(X) for discrete distributions

Answer 1

E(X) = sum of X P(X)

Question 2

For discrete distributions, the cumulative distribution function =

Answer 2

CDF = sum of PDF

Question 3

For continuous distributions, the cumulative distribution function =

Answer 3

CDF = integral of PDF

Question 4

E(X + a) =

Answer 4

E(X) + a

Question 5

E(aX) =

Answer 5

a E(X)

Question 6

V(X) formula

Answer 6

V(X) = E[(X - E(X))^2] = E(X^2) - [E(X)]^2

Question 7

V(X + a) =

Answer 7

V(X) - adding a constant doesn’t affect variance

Question 8

V(aX) =

Answer 8

a^2 V(X) since variance is a squared operator

Question 9

E(X + Y) =

Answer 9

E(X) + E(Y)

Question 10

V(X + Y) =

Answer 10

V(X) + V(Y) + 2COV(X, Y)

Question 11

COV(X, Y) =

Answer 11

E(XY) - E(X)E(Y)

Question 12

V(X - Y) =

Answer 12

V(X) + V(Y) - 2COV(X, Y)

Question 13

V(aX - bY) =

Answer 13

a^2 V(X) + b^2 V(Y) - 2ab COV(X, Y)

Question 14

What is the covariance if X & Y are independent?

Answer 14

COV(X, Y) = 0

Question 15

COV(X + a, Y) =

Answer 15

COV(X, Y)

Question 16

COV(aX, Y) =

Answer 16

aCOV(X, Y)

Question 17

Conditional probabilities must sum to…

Answer 17

1

Question 18

How to work out probability between an interval for continuous distributions

Answer 18

Integrate f(X) dX between the interval [a, b]

Question 19

Variance for a continuous distribution =

Answer 19

Integral of X^2 f(X) dX - [integral of X f(X) dX]^2

Question 20

What is the expectation of the sample mean when we have 3 drawings?

Answer 20

E(X1 + X2 + X3/3) = 1/3 [E(X1) + E(X2) + E(X3)] = 1/3 x 3M = M

Question 21

What is the variance of a sample mean when we have 3 drawings?

Answer 21

V(X1 + X2 + X3/3) = 1/9 [V(X1) + V(X2) + V(X3)] = 3sigma^2/9 = sigma^2 / 3

Covariance zero as independent random samples