Exam tomorrow and some quick review

Got an exam tomorrow for statistics and I should be going back to study. This is on Chapter 6,7 and 8 and it will cover things like Methods of Distribution function, Order Statistics, Sampling distribution and Central limit theorem, Estimation.

Some definitions
Estimation
E(theta_hat) = theta then we call it an unbiased estimator
Order Statistics
This comes into play when we are talking about ordering of Random variables. For example if we have random variables like Y1, Y2,>.. Yn and suppose Y3 can take the largest number of values then Y3 is the max and it is denoted by Y(n), similarly now you can guess the smallest random variable will be christened as Y(1) and all others in between.
Distribution function
From a given Distribution function we can generate different functions and here the method of distributions come in handy for example. There are mainly 3 different types of method including
1. Distribution function method
2. Transformation function method
3. Method of moment generating functions

There is one more called Multivariable Transformations Using Jacobians
I would like to comment on the method of moment generating method.
Moment generating function is E(exp^(Y*t)) which has close relation to E(Y)
By differentiating m(t) = E(exp^(Y*t)) and substituting t = 0 one can get different value of E(Y^n). Thus to find the value of E(Y^3) all one needs to do is differentiate m(t), 3 times and substitute t = 0. With moment's one can easily find Variance which is usually describe as
V(Y) = E(Y^2) - (E(Y))^2
all one needs to do is find the moment of the function.
Some common moments should be remember
like for
Poisson Distribution exp(lambda*(exp(t)-1))
Normal Distribution exp(mu*t+ (t^2+sigma^2)/2)
GammaDistribution 1/(1-Beta*t)^(sigma)
Binomial Distribution (p*exp(t)+q)^n
it helps to remember and recognize these distributions. Because they are not only easy to remember they can also help in finding out other distributions.
For example the moment of Z = Y-mu is exp(t^2*sigma^2/2) and the moment of Z = (Y-mu)/sigma is exp(t^2/2). One very important result to know and derive is that moment of Z^2 is a Chi-Square distribution.
A note on Gamma Distribution should be interesting here as there are two more distributions Exponential and Chi square are speical case of Gamma Distribution. The Beta distribution is different than Gamma Distribution and it moment doesn't exist in close form.