Distributions derived from gaussian

Question 1

Define chi squared RV

Answer 1

A RV X is chi-squared with ν > 0 degrees of freedom iff X ∼ Ga (ν/2, 1/2)

Question 2

What is the transformation to a standard normal RV to get a chi squared

Answer 2

The square of the standard normal RV gives chi squared

Question 3

What distribution does a chi squared distribution have a relation with

Answer 3

Its a special case of gamma distribution

Question 4

What does the mean and variance of an estimator tell us

Answer 4

Info on how well we are estimating the parameter of interest:

• The mean tells us if θ_estimate is centred around θ.
• The variance measures the uncertainty

Question 5

What can be a problem with the sampling distribution of an estimator

Answer 5

However, since the sampling distribution of ˆθ may depend on θ, also its
moments may involve θ, which is unknown. So the uncertainty associated to an estimate for the sample mean may contain a variance or parameter of the population which is unknown.

Question 6

What is the problem with the normal sample mean

Answer 6

It is Xbar is normally distributed and unbiased but the uncertainty of Xbar involves the variance of the population which is unknown.

Question 7

Why do we estimate T - from student T distribution

Answer 7

Note: in a Gaussian random sample, the variance of T = Xbar −μ/ √S2/n does not
depend on σ.

Question 8

What is the mgf of a student t distribtuion and why

Answer 8

Student’s t has no mgf because it does not have moments of all orders
(i.e. some moments are not finite)

Question 9

What is significant about the moments of the student t distribution

Answer 9

If there are n − 1 = p degrees of freedom, the moments only up to
(including) degree p− 1 exist
So t1 has no mean and t2 has no variance

Question 10

What is t1 distribution called

Answer 10

Cauchys distribution

Question 11

What estimators can use cramer rao’s inequality

Answer 11

Unbiased estimators it will hold

Question 12

How we can we tell a best unbiased estimator and what is it called

Answer 12

Consequence: the best unbiased estimator is the one attaining this lower
bound, the cramer rao bound.
This special estimator is called Minimum Variance Unbiased Estimator
(MVUE).

Question 13

What should we do with all unbiased estimators

Answer 13

Whenever we have an unbiased estimator we should check if it attains the
CRLB

Question 14

State cramer rao inequality

Answer 14

Let X1, . . . , Xn
IID ∼ fX (·|θ), and let us have an unbiased estimator of θ. Then,
under smoothness assumptions on Lx (θ), the following holds: Var(estimator)
≥ 1/ E [I (θ)]