Functions Of Variables

Question 1

Give the name for random variables which are independent and all have the same distribution function

Answer 1

Independent and identically distributed random variables; I.I.D

Question 2

Define the n-th partial sum of a sequence of random variables

Answer 2

Question 3

State the distribution of the n-th partial sum of IID Bernoulli (p) random variables

Answer 3

S_n will have a Bin(n,p) distribution

Question 4

State the distribution of the n-th partial sum of IID Exp (λ) random variables

Answer 4

S_n will have a Gamma (n,λ) distribution

Question 5

State the distribution of the n-th partial sum of IID Poi (λ) random variables

Answer 5

S_n will have a Poi(nλ) distribution

Question 6

State the distribution of the n-th partial sum of IID N(µ, σ²) random variables

Answer 6

S_n will have a N(nµ, nσ² ) distribution

Question 7

Describe how to work out the expected value of S_n

Answer 7

Question 8

Describe how to work out the variance of S_n

Answer 8

Question 9

Define the central limit theorem

Answer 9

Question 10

Give the formula to transform random variables where X is crv with pdf fx and Y=g(X)

Answer 10

Note: the formula can be rewritten in terms of g where h = g^-1

This formula can only be used with monotonic functions, ie those which are either nonincreasing or nondecreasing.

Question 11

Describe the MGF of the sum, S_n

Answer 11

The MGF of S_n is the product of the MGFs of the IIDs.