FS1 Chapters 1-5

Question 1

How do you find E(X) for discrete random variables?

Answer 1

ΣxP(X=x)
Multiply each outcome and probability together and add

Question 2

How do you find E(X^2) for discrete random variables?

Answer 2

Σx^2P(X=x)

Question 3

How do you find Var(X) for discrete random variables?

Answer 3

E(X^2) - (E(X))^2
Mean of the squares minus square of the mean

Question 4

What is E(aX + b) equal to?

Answer 4

aE(X) + b

Question 5

What is E(X+Y) equal to?

Answer 5

E(X) + E(Y)

Question 6

What is Var(aX + b) equal to?

Answer 6

a^2Var(X)

Question 7

What are the conditions required to use the Poisson distribution?

Answer 7

-Events are independent
-Events occur singly in space or time
-Constant average rate so mean number in a period is proportional to length of period

Question 8

What is the combined distribution of X and Y, X~ Po(λ) and Y~ Po(μ)?

Answer 8

X+Y~ Po(λ+μ)

Question 9

What is E(X) and Var(X) in the Poisson distribution?

Answer 9

E(X)= λ
Var(X)= λ

Question 10

What is E(X) of a binomial distribution X~ B(n,p)?

Answer 10

E(X) = np

Question 11

What is Var(X) of a binomial distribution X~ B(n,p)?

Answer 11

Var(X) = np(1-p)

Question 12

What are the conditions needed to use a binomial distribution as an approximation for a Poisson distribution?

Answer 12

n is large
p is small
λ = np

Question 13

What is E(X) of a distribution X~ Geo(p)?

Answer 13

E(X)= 1/p

Question 14

What is Var(X) of a distribution X~ Geo(p)?

Answer 14

Var(X)= (1-p)/p^2

Question 15

What is E(X) of a distribution X~ NB(r,p)?

Answer 15

E(X) = r/p

Question 16

What is Var(X) of a distribution X~ NB(r,p)?

Answer 16

Var(X)= r(1-p)/ p^2

Question 17

What is the central limit theorem?

Answer 17

The mean of a large random sample taken from any random distribution is always approximately normally distributed

Question 18

What is the distribution for the central limit theorem?

Answer 18

For distribution with mean μ and standard deviation σ and sample size of n ,
Xbar ~ N(μ, (σ/√n)^2)