Probabilities
Robert B. Grifths
Version oF 12 January 2010
ReFerences:
±eller,
An introduction to probability theory and its applications
, Vol. 1, 3d ed (Wiley 1968). See Intro-
duction, Ch. I, Ch. V
DeGroot and Schervish,
Probability and Statistics
, 3d ed (Addison-Wesley, 2002), Chs 1, 2, 3, 4
Very compact introductions to material relevant to quantum mechanics:
QCQI =
Quantum Computation and Quantum Information
by Nielsen and Chuang (Cambridge, 2000),
App. 1
CQT =
Consistent Quantum Theory
by Robert Grifths (Cambridge, 2002), Secs. 5.1, 8.2, 9.1, 9.2
Contents
1B
a
s
i
c
s
1
1.1
Sample space .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .....
1
1.2
Event algebra .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .......
2
1.3
Probabilities .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .........
2
1.4
Ensembles .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .....
3
1.5
Conditional probabilities .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .........
3
2R
a
n
d
omV
a
r
i
a
b
l
e
s
3
2.1
Random variables, averages, indicator Functions .
. . . . . . . . . . . . . . . . .........
3
2.2
Probability distributions .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . ..........
4
2.3
Joint and conditional distributions .
. . . . . . . . . . . . . . . . . . . . . . . .........
4
2.4
Independent random variables .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....
5
2.5
Variance, covariance, correlation .
. . . . . . . . . . . . . . . . . . . . . . . . .........
6
3 Stochastic Processes
6
3.1
Examples; sample spaces .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .......
6
3.2
Probabilities .
7
3.3
Markov chains
7
1
Basics
±
Ordinary (classical) probability theory uses three key concepts: sample space
S
, event algebra
B
,
probabilities
P
.
1.1
Sample space
±
Sample space
S
oF
mutually exclusive possibilities
, thought oF as outcomes oF an ideal experiment, one
and only one oF which actually occurs, or is true, in a particular case.
◦
Examples. Coin toss:
H
or
T
. Die:
{
s
=1
,
2
,
3
,
4
,
5
,
6
}
◦
A sample space can be either discrete or continuous (e.g., integers, real numbers), and in the Former
case either ²nite or in²nite. ±or our purposes it sufces to consider ²nite discrete sample spaces.
1