Special Distributions

Question 1

Describe Bernoulli trial

Answer 1

Discrete distribution with 2 outcomes. X=1 is success and X=0 is failure. Special case of binomial where n=1.

Question 2

E(X) & V(X) for Bernoulli

Answer 2

E(X) = P
V(X) = P(1 - P)

Question 3

What is a binomial distribution?

Answer 3

Repeated applications of Bernoulli trial, where each individual has the same probability. It describes the number of success in n trials.

Question 4

Formula for binomial distribution

Answer 4

X ~ B(n, p(success))

Question 5

What is P(X) for binomial

Answer 5

P(X) = nCr x P^X x (1 - P)^(n - x)

Question 6

E(X) and V(X) for binomial

Answer 6

E(X) = np
V(X) = np(1-p)

Question 7

What is a poison distribution? Discrete or continuous?

Answer 7

Number of occurrences of an event in a defined time interval. Discrete distribution.

Question 8

What do we need for a Poisson distribution?

Answer 8

N to be large

Events in each period to be independent

Question 9

What is E(X) and V(X) for poisson?

Answer 9

E(X) = V(X) = lambda ‘mean rate of occurrence / success in each interval’

Question 10

What can we use as an approx to binomial? When?

Answer 10

Use poisson to approx binomial
Where n >50
P small p<0.1

Question 11

Notation for uniform distribution

Answer 11

X ~ U(a, b)

Question 12

What type of distribution is the uniform distribution?

Answer 12

Most basic continuous distribution, all values of X have an equal probability. Symmetric.

Question 13

What is the PDF for uniform distribution?

Answer 13

f(X) = 1 / (b - a)

Question 14

E(X) & V(X) for uniform distribution

Answer 14

E(X) = (b + a) / 2

V(X) = (b - a)^2 / 12

Question 15

How to work out probabilities for uniform distribution

Answer 15

Integrate 1 / b - a w.r.t. X between the interval c and d

Or P(c < X < d) = d - c / b - a

Question 16

Notation for normal distribution

Answer 16

X ~ N (M, sigma^2)

Question 17

What values can X take for normal distribution

Answer 17

Between +- infinity

Question 18

Relation between mean, median & mode for normal distribution

Answer 18

Mean = median = mode

Question 19

Standard normal formula

Answer 19

Z = X - M / sqrt sigma^2

Question 20

If X1 and X2 are normally distributed, what distribution does a linear combination of them follow?

Answer 20

Any linear combination of two normally distributed variables is also normal.

X1 + X2 ~ N(M1 + M2, sigma1^2 + sigma2^2) covariance are zero is independent random samples.

Question 21

Possible range of values of X for chi squared

Answer 21

Only positive X values due to squaring

Question 22

E(X) & V(X) for chi squared

Answer 22

E(X) = V (DOF)
V(X) = 2V

Question 23

Relation between chi squared and normal distribution

Answer 23

Z^2 = N(0, 1)^2 = chi squared 1

Chi squared with DOF = 1 is the square of the standard normal distribution

Question 24

If we sum up chi squared, what is the new DOF?

Answer 24

DOF of linear combination = sum of individual DOF

Question 25

Formula for F distribution

Answer 25

Fm,n = (chi squared m/m) / (chi squared n/n)

Question 26

Relation between F and chi squared

Answer 26

F = ratio of 2 independent chi squared distributions scaled by their DOFs

Question 27

E(X) & V(X) for F distribution

Answer 27

E(X) = n/(n-2)
V(X) = 2n^2 (m + n - 2) / m(n - 2)^2 (n - 4)

Question 28

As n tends towards infinity (DOF of denominator), what happens to F distribution?

Answer 28

As n to infinity, F collapses onto a single chi squared m / m (the numerator)

Question 29

Formula for t distribution

Answer 29

T ~ N(0, 1) / sqrt chi squared n / n

Question 30

What happens to a t distribution as n tends towards infinity?

Answer 30

T collapses onto N(0, 1) I.e. Standard normal distribution

Question 31

E(X) and V(X) for t distribution

Answer 31

E(X) = 0
V(X) = n / (n - 2)

Question 32

Relation between t distribution and F distribution

Answer 32

Square of t = F distribution when M=1 (when numerator of F is a chi squared 1)