Random Variables and Probability
Distributions
August 26, 2006
1
1.1
Random Variables
Denition of a Random Variable
Consider an experiment with a sample space . Then consider a function
X that assigns to each possible outcome in an experiment (! 2 ) one a
BASIC STATISTICS
1. S AMPLES , R ANDOM S AMPLING AND S AMPLE S TATISTICS
1.1. Random Sample. The random variables X1 ; X2 ; :; Xn are called a random sample of size n
from the population f (x) if X1 ; X2 ; :; Xn are mutually independent random variables a
MULTIVARIATE PROBABILITY DISTRIBUTIONS
1. P RELIMINARIES 1.1. Example. Consider an experiment that consists of tossing a die and a coin at the same time. We can consider a number of random variables dened on this sample space. We will assign an indicator
INTERVAL ESTIMATION AND HYPOTHESES TESTING
1. B RIEF R EVIEW OF S OME C OMMON D ISTRIBUTIONS
1.1. Probability Distributions. The table below contains information on some common probability distributions.
Distribution
Normal, N
;
pdf
2
1
Exponential
Chi-sq
1
A Review of Set Notation
1.1
Denition of a Set
A set is any collection of objects which are called its elements. If x is an element of the set
S, we say that x belongs to S and write
x 2 S:
If y does not belong to S, we write
y 62 S:
The simplest way to
SAMPLE MOMENTS
1. P OPULATION M OMENTS
1.1. Moments about the origin (raw moments). Recall that the rth moment about the origin of a random
variable X, denoted by 0 , is the expected value of X r ; symbolically,
r
X
0
r
xr f (x)
r = E(X ) =
x
for r = 0; 1
TRANSFORMATIONS OF RANDOM VARIABLES
1. I NTRODUCTION
1.1. Denition. We are often interested in the probability distributions or densities of functions of
one or more random variables. Suppose we have a set of random variables, X1 , X2 , X3 , . . . Xn ,
wi
SOME SPECIFIC PROBABILITY DISTRIBUTIONS
1. N ORMAL RANDOM VARIABLES
1.1. Probability Density Function. The random variable X is said to be normally distributed with mean
and variance 2 (abbreviated by x N [ ; 2 ] if the density function of x is given by
2
SOME THEOREMS ON QUADRATIC FORMS AND NORMAL VARIABLES
1. T HE M ULTIVARIATE N ORMAL D ISTRIBUTION The n 1 vector of random variables, y, is said to be distributed as a multivariate normal with mean vector and variance covariance matrix (denoted y N ( ; )