Lecture 7 Notes: Linearized Error Propagation (Part 1)
Last time: Moments of the Poisson distribution from its generating function.
G(s) e ( s1)
dG
( s1)
ds
d 2G
( s1)
2
ds
2
X
s1
X2
s1
s1
2
2 X 2 X 2
2
2
X
Example: Using telescope to measure i

Lecture 2 Notes: Independence
Last time:
Given a set of events with are mutually exclusive and equally likely,
n(E)
P(E)
.
Example: card games
Number of different 5-card poker
52
hands
2,598,960
Number of different 13-card bridge
52
hands
013,559,6

Lecture 4 Notes: Correlation, Covariance, and Orthogonality
Last time: Left off with characteristic function.
4. Prove x (t) x (t) where X X 1 X 2 . X n (Xi
i
independent) Let S X 1 X 2 .X n where the Xi are independent.
s (t ) E e jtS E e jt X X . X
E e

Lecture 5 Notes: Some Common Applications of Linear Dependence
Last time:
Characterizing groups of random variables
Names for groups of random variables
S
X
i
i1
S2
n
n
X iX j
i1 j1
Characterize by pairs to compute
E XY XY
dx xyf
, y )dy
x, y
(x
which we

Lecture 3 Notes: Expectation, Averages and Characteristic Function
Last time: Use of Bayes rule to find the probability of each outcome in a set of
P( A | E1 , E 2 ,. En )
In the special case when Es are conditionally independent (though they all
depend o

Lecture 6 Notes: Multivariate Normal Density Function
Example: Sum of two independent random variables
Z=X+Y
f z (z)dz P(a Z b)
b
a
P(a X Y b)
P(a X Y b X )
lim P( x X x dx)P(a x Y
b x) dx x
bx
lim f ( x) dx
( y)dy dx 0 x
f x (x) dx
bx
a x
f
a
f y ( y)dy

Lecture 8 Notes: Linearized Error Propagation (Part 2)
Last time: Multi-dimensional normal distribution
exp x x T M 1 x x
f (x )
1 M
2
2 2
If a set of random variables Xi having the multidimensional normal distribution
is uncorrelated (the covariance ma

Lecture 10 Notes: Autocorrelation Function
Last time: Random Processes
With f (x,t) we can compute all of the usual statistics.
Mean value:
x(t1 )
xf
(x, t1 )dx
Mean squared value:
x(t1 )2
x
2
f
(x, t1 )dx
Higher order distribution and density functions

Lecture 9 Notes: Stationarity, Ergodicity, and Classification of
Processes
Last time: Linearized error propagation
e Se
Integrate the errors at deployment to find the error at the surface.
Es e e T
S e e T
ST
SE1S T
Or can be integrated from:
F, where (0)

Lecture 1 Notes: Random Signals
Pre-requisites
6.041 Probabilistic systems
Summary of the subject (topics)
1. Brief review of probability
a. Example applications
2. Brief review of random variables
a. Example applications
3. Brief review of random process