Axioms of Probability
- where is your sample space
- For mutually exclusive events ,
- For all events , ,
Expectation and Variance
Some definitions and properties.
Let for some distribution .
Let be the support or domain of your distribution.
Dr. Xu refers to this as the smooth property.
This one is not used as often on tests as total expectation
Moments and Moment Generating Functions
We call the i'th moment of .
We call the i'th central moment of .
Therefore the mean is the first moment and the variance is the second central moment.
Moment Generating Functions
We call this the moment generating function (mgf).
We can differentiate it with respect to and set to get the higher moments.
- The mgf, if it exists, uniquely defines the distribution.
- The mgf of is
There are 4 types of convergence typically taught in undergraduate courses.
See Wikipedia Convergence of random variables
- Implies Convergence in distribution
Pointwise convergence of the cdf
A sequence of random variables converges to in probability
if for all ,
- Equivalent to convergence in probability if it converges to a degenerate distribution (i.e. a number)
In Mean Squared
Let be a function such that exists and
- You can think of this like the Mean Value theorem for random variables.
Inequalities and Limit Theorems
Let be a non-negative random variable.
Apply Markov's inequality:
- Usually used to prove convergence in probability
Central Limit Theorem
Very very important. Never forget this.
For any distribution, the sample mean converges in distribution to normal.
Different ways of saying the same thing:
Law of Large Numbers
The sample mean converges to the population mean in probability.
For all ,
- The sample mean converges to the population mean almost surely.
Relationships between distributions
This is important for tests.
See Relationships among probability distributions.
Sum of poission is poisson sum of lambda.
- If and then for any
Note exponential distributions are also Gamma distrubitions
- If then .
- If and then .
- If and , then .
Ratio of normal and squared-root of Chi-sq distribution yields T-distribution.
The ratio of two normalized Chi-sq is an F-distributions
Too many. See the Wikipedia Page.
Most important are Chi-sq and T distribution