1. Let X1 , … , Xn be a random sample from a distribution with mean μ and variance σ2. Find the mean and variance of the sample mean

2. Let X1 , … , Xn be a random sample from a distribution with mean μ and variance σ2. Find the mean of

3. Let X1 , … , Xn be a random sample from N (μ ,σ2 ) distribution. Find the mean and variance of

4. State the following Theorems:

- Law of Large Numbers

- Central Limit Theorem

- Student’s Theorem

- Cramer – Rao Theorem.

5. Let X1 , … , Xn be a random sample from N (μ ,σ2 ) distribution. Define

6. For a random sample from given distribution

a) find the method of moment estimator of the unknown parameter,

b) find the mle of the unknown parameter,

c) check the cosistency of method of moments estimators.

7. For a rv X with pdf (pmf) f(x, θ), define I(θ) – the Fisher Information and state both formulas: using first and second derivative of ln f(x,θ).
8. For a random sample from given distribution with unknown parameter θ, find the Fisher Information in the sample and then CRLB for the variances of unbiased estimators of θ. Also

a) for a given estimator of θ, check if it is an unbiased estimator of θ; check the efficiency of this estimator,

b) find an efficient estimator of θ.

9. Confidence Intervals: in each case find (derive) the (1 - α)100% (or 90%, 95%, 99%) confidence interval. Indicate (or read from Tables the appropriate quantiles).

a) If a random sample is from N(μ ,σ2 ):

- For the mean when the variance is known,

- For the mean when the variance is unknown,

- For the variance when the mean is known,

- For the variance when the mean is unknown,

- In case of two samples from normal distribution: for the difference of two means when variances are known.

Note that you will be deriving exact confidence intervals.

b) If a random sample is from (any) distribution with finite variance:

- For the mean when the variance is known,

- For the mean when the variance is unknown.

- In case of two samples, for the difference of two means when  variances are known and also when variances are unknown.

Note that you will be deriving approximate confidence intervals (assuming that n is large).

c) If a random sample is from Bernoulli (p) distribution:

- For the parameter p

- In case of two samples, for p1 – p2 .

Note that you will be deriving approximate confidence intervals (assuming that n is large).

d) If a random sample is from gamma distribution (or exponential, which is a special case of gamma distribution):

- For the mean of exponential distribution

- For the second parameter of gamma distribution.

Note that you will be deriving exact confidence intervals.

Remark: Always develop with justification pivotal quantity and use it to find (derive) a confidence interval.