Properties of Random Samples

Question 1

Define a random sample

Answer 1

Random variables X1,….Xn are called a random sample of size n from population fX (x|θ) if they are mutually independent and the marginal pdf or pmf of each Xi is the same function fX (x|θ)

Question 2

What are we assuming in a random sample

Answer 2

We are assuming that each RV are independent and that they are
observed under the same conditions

Question 3

Define an estimator

Answer 3

A statistic (or estimator) is any function T (X1, . . . , Xn) of a random sample. T is a Random variable

Question 4

What is the distribution of an estimator called - (estimator is an RV)

Answer 4

Sampling distribution

Question 5

Describe the difference between the sample mean and sample variance and the mean and variance

Answer 5

The sample mean and variance are RVs, whereas the mean and variance are
moments associated to a RV. It would make perfect sense to evaluate, for example, the “mean
of the sample mean”, the “variance of the sample mean”

Question 6

What symbol denotes and estimator

Question 7

Define an unbiased estimator

Answer 7

an estimator is unbiased if the expectation of the sampling

distribution is equal to the parameter of interest

Question 8

What is the problem with Ψ^2 as an estimator for variance and how can it be corrected

Answer 8

It is a biased estimator but Ψ^2 tends to be unbiased as n → ∞.
Its asymptotically unbiased

Question 9

Why do we used Ψ^2 instead of S^2 as an estimator

Answer 9

Even though its biased it gives a smaller variance than that of s^2

Question 10

Let X1, . . . , Xn be a random sample from a Gaussian population with mean μ
and variance σ^2. What is significant about the estimators for the mean and variance

Answer 10

Xbar and S^2 are independent RVs