Chapter 1 Sample Space and Probability
1 Sets
S T = cfw_x; x S or x T and S T = cfw_x; x S and x T If S = cfw_, , and T = cfw_, , then S T = cfw_, , , and S T = cfw_, . Shade on separate graphs S T and S T . S S
T
T
Let S be the set of (strictly) p
Chapter 2 Discrete Random Variables
1 Basic Concepts
A random variable is a real-valued function of the outcome of the experiment. The number of heads out of two tosses of a coin denes a function (mapping) from the sample space = cfw_HH, HT, T H, T T in
Chapter 3 General Random Variables
1 Continuous Random Variables and PDFs
A random variable X is continuous if there is a nonnegative fX , called the probability density function of X (PDF), such that
b
P[a X b] = If X is a continuous RV with PDF fX , th
Chapter 4 Further Topics on Random Variables
1 Transforms
The moment generating function (MGF) of a random variable X is MX (t) = E[etX ]. In the discrete case, M X (t) =
x
etx pX (x).
In the continuous case, M X (t) =
etx fX (x)dx.
Note that in general
Chapter 5 Limit Theorems
Let X1 , . . . , Xn be independent and identically distributed random variables with mean 1 and variance 2 . If Mn = (X1 + . . . + Xn ) then n E[Mn ] = and var(Mn ) = 2 . n
As n +, var(Mn ) 0 and Mn approaches (in some sense). T