FS1 Chapters 1-8 Flashcards
How do you find E(X) for discrete random variables?
ΣxP(X=x)
Multiply each outcome and probability together and add
How do you find E(X^2) for discrete random variables?
Σx^2P(X=x)
How do you find Var(X) for discrete random variables?
E(X^2) - (E(X))^2
Mean of the squares minus square of the mean
What is E(aX + b) equal to?
aE(X) + b
What is E(X+Y) equal to?
E(X) + E(Y)
What is Var(aX + b) equal to?
a^2Var(X)
What are the conditions required to use the Poisson distribution?
-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
What is the combined distribution of X and Y, X~ Po(λ) and Y~ Po(μ)?
X+Y~ Po(λ+μ)
What is E(X) and Var(X) in the Poisson distribution?
E(X)= λ
Var(X)= λ
What is E(X) of a binomial distribution X~ B(n,p)?
E(X) = np
What is Var(X) of a binomial distribution X~ B(n,p)?
Var(X) = np(1-p)
What are the conditions needed to use a binomial distribution as an approximation for a Poisson distribution?
n is large
p is small
λ = np
What is E(X) of a distribution X~ Geo(p)?
E(X)= 1/p
What is Var(X) of a distribution X~ Geo(p)?
Var(X)= (1-p)/p^2
What is E(X) of a distribution X~ NB(r,p)?
E(X) = r/p
What is Var(X) of a distribution X~ NB(r,p)?
Var(X)= r(1-p)/ p^2
What is the central limit theorem?
The mean of a large random sample taken from any random distribution is always approximately normally distributed
What is the distribution for the central limit theorem?
For distribution with mean μ and standard deviation σ and sample size of n ,
Xbar ~ N(μ, (σ/√n)^2)
What are the hypotheses for chi squared tests?
Ho: observed values follow theoretical distribution
H1: observed values do not follow theoretical distribution
How is the chi squared value calculated?
Σ(O-E)^2/E
O: observed values
E: expected values
When do you combine classes?
When expected values are less than 5, you combine down until they are greater than 5
How do you find the number of degrees of freedom usually?
Number of classes (after combining) - 1
When is the result significant for chi squared tests?
When the chi squared value is greater than the critical value, so Ho is rejected
What do you do when you are given or not given the parameter (probability or mean) for a chi squared test?
If given, do nothing
If calculated, - 1 from degrees of freedom
How do you calculated expected frequency for contingency tables?
(row total x column total)/grand total
How do you calculate degrees of freedom for a contingency table?
(no. rows - 1) x (no. columns - 1)
What do you do if expected frequency in a column is less than 5?
Combine columns
What is a pgf and when can it be used?
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
How is a pgf given?
Gx(t) = P(X=x)t^x, where t is a dummy variable
Gx(t) = E(t^x)
What is Gx(1) equal to for any pgf?
1
How is the mean of a distribution found using a pgf?
E(X) = G’x(1)
How is the variance of a distribution found using a pgf?
Var(X) = G’‘x(1) + G’x(1) - (G’x(1))^2
If you have a pgf for two variables X and Y, what is the pgf for Z = X + Y?
Gz(t) = Gx(t) x Gy(t)
If you have a pgf for X, what is the pgf for Y = aX + b?
Gy(t) = t^b Gx(t^a)
What is the probability of a type I error?
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
What is the probability a type II error?
Probability of wrongly accepting Ho, so probability of NOT being in the critical region
What is the probability of a type I error for a continuous distribution?
For continuous distributions, eg normal distribution P(Type I error) = significance level of test
What is the relationship between the probability of type I and type II errors?
They are inversely proportional
How can the probability of type I and type II errors be decreased?
Increase sample size n
What is the size of a test?
Probability of wrongly rejecting Ho, so probability of being in the critical region, same as P(type I error)
What is the power of a test?
Probability of correctly rejecting Ho, 1- P(type II error)
1- probability of not being in critical region
What is the power function of a test?
A function of the parameter p which gives the probability that Ho will be rejected if p is the true value of the parameter
When comparing tests of similar sizes, what is desirable?
A higher power within the likely range of the parameter
