FS1 Chapters 1-8

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)

Question 19

What are the hypotheses for chi squared tests?

Answer 19

Ho: observed values follow theoretical distribution
H1: observed values do not follow theoretical distribution

Question 20

How is the chi squared value calculated?

Answer 20

Σ(O-E)^2/E
O: observed values
E: expected values

Question 21

When do you combine classes?

Answer 21

When expected values are less than 5, you combine down until they are greater than 5

Question 22

How do you find the number of degrees of freedom usually?

Answer 22

Number of classes (after combining) - 1

Question 23

When is the result significant for chi squared tests?

Answer 23

When the chi squared value is greater than the critical value, so Ho is rejected

Question 24

What do you do when you are given or not given the parameter (probability or mean) for a chi squared test?

Answer 24

If given, do nothing
If calculated, - 1 from degrees of freedom

Question 25

How do you calculated expected frequency for contingency tables?

Answer 25

(row total x column total)/grand total

Question 26

How do you calculate degrees of freedom for a contingency table?

Answer 26

(no. rows - 1) x (no. columns - 1)

Question 27

What do you do if expected frequency in a column is less than 5?

Answer 27

Combine columns

Question 28

What is a pgf and when can it be used?

Answer 28

A probability generating function is a function that stores details of a probability distribution. It can be used with a discrete probability distribution that takes non-negative integer values

Question 29

How is a pgf given?

Answer 29

Gx(t) = P(X=x)t^x, where t is a dummy variable Gx(t) = E(t^x)

Question 30

What is Gx(1) equal to for any pgf?

Answer 30

1

Question 31

How is the mean of a distribution found using a pgf?

Answer 31

E(X) = G'x(1)

Question 32

How is the variance of a distribution found using a pgf?

Answer 32

Var(X) = G''x(1) + G'x(1) - (G'x(1))^2

Question 33

If you have a pgf for two variables X and Y, what is the pgf for Z = X + Y?

Answer 33

Gz(t) = Gx(t) x Gy(t)

Question 34

If you have a pgf for X, what is the pgf for Y = aX + b?

Answer 34

Gy(t) = t^b Gx(t^a)

Question 35

What is the probability of a type I error?

Answer 35

Probability of wrongly rejecting Ho, so probability of being in the critical region. It is the same as the actual significance level for a hypothesis test

Question 36

What is the probability a type II error?

Answer 36

Probability of wrongly accepting Ho, so probability of NOT being in the critical region

Question 37

What is the probability of a type I error for a continuous distribution?

Answer 37

For continuous distributions, eg normal distribution P(Type I error) = significance level of test

Question 38

What is the relationship between the probability of type I and type II errors?

Answer 38

They are inversely proportional

Question 39

How can the probability of type I and type II errors be decreased?

Answer 39

Increase sample size n

Question 40

What is the size of a test?

Answer 40

Probability of wrongly rejecting Ho, so probability of being in the critical region, same as P(type I error)

Question 41

What is the power of a test?

Answer 41

Probability of correctly rejecting Ho, 1- P(type II error) 1- probability of not being in critical region

Question 42

What is the power function of a test?

Answer 42

A function of the parameter p which gives the probability that Ho will be rejected if p is the true value of the parameter

Question 43

When comparing tests of similar sizes, what is desirable?

Answer 43

A higher power within the likely range of the parameter