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332:421 Communications Engineering
Lecture Notes
9-11-2000
Probability Review
Experiment
= Procedure + Observation
Sample space (S) = mutually exclusive, collectively
exhaustive, finest grain set of all possible outcomes.
Random Variable (X): is a function that maps an element of
the sample space s Є S to R
Random Process is a mapping of the sample outcome s Є S to
a waveform X(t, s)
Example: roll a die. Result of the roll is N. N Є {1,...,6}
If N = n, then let X(t,n) = Xn(t) = { n
0
t ≥ 0
o.w.
roll a die an infinite number of times. We generate an
infinite sequence
S = [n1, n2, n3, …]
S = [1,3,4,6,1,1,6,…]
7
6
5
4
X(t )
3
2
1
0
0
1
2
3
4
5
6
7
8
Map S to the waveform X(t, s) such that:
X(t) = { 0 t ≥ 0
nk k-1 < t ≤ k, k = 1,2,3,……
So knowing X(t0) may or may not help you predict X() when
t1 ≠ t0
if you know the sample outcome, you know everything.
Description:
If we have a random variable X; with describe it with a
cumulative distribution function:
FX(x) = P[X ≤ x]
Probability Density Function
fX(x) = dfX(x)/dx
P[x < X ≤ x + dx] = fX(x) dx
If an experiment produces multiple random variables
S → X1, X2,..,Xn
Joint c.d.f
FX1(x1xn - xn) = P[X1 ≤ x1, X2 ≤ x2, X3 ≤ x3,…, Xn ≤ xn]
Joint p.d.f
fX1...xn(x1...xn) = δfX1...xn(x1...xn)/δx1...δxn
P[x1 ≤ X1 ≤ y1, x2 ≤ X2 ≤ y2,..., xn ≤ Xn ≤ yn]
= ∫...∫ fX1...xn(x1...xn)δx1...δxn
X1...Xn = n r.v’s (random vectors)
FX(x) = FX1...xn(x1...xn)
Random Process X(t) sampled at time instants
X = [X(t1),
X(t2),..., X(tk)]
Random Process Model: given t1...tk, a model tells you the
joint CDF or PDF
Roll a die an infinite number of times
X(t1) = the result of roll 11. equally likely to be 1,2,..,6
Given, t1 = 11.1 and t2 = 12.2
fX(t1)X(t2)(x1,x2) = P[X(t1) ≤ x1, X(t2) ≤ x2]
= P[X(t1) ≤ x1] P[X(t2) ≤ x2]
= FX(t1)(x1) FX(t2)(x2)
Given, t1 = 11.1 and t2 = 11.3
fX(t1)X(t2)(x1,x2) = P[roll 11 ≤ x1, roll 11 ≤ x2]
= P[roll 11 ≤ min(x1, x2)]
stationarity
When is a random process independent of time?
X(t) is a stationary random process if given any time
instants t1...tk and time offsets T, then
fX(t1)...x(tk)(x1...xk) = fX(t1+T)...X(tk+T)(x1...xk)
at k = 1,
fX(t1)(x1)
= fX(t1+T)(x1)
= fX(0)(x1)
= fX(x1)
for all T
at k = 2,
fX1(t1)X2(t2)(x1,x2) = fX(t1+T)X(t2+T)(x1,x2)
substituting:
t2 = t1 + τ and T = -t1
therefore:
fX(t1)X(t1+τ)(x1,x2) = fX(0)X(τ)(x1,x2)
X(t,s) is a random process
Sample function So
X(t,So) = xo(t)
←
some random output waveform
Time average:
xo(t) = limT
→ ∞
1/T∫xo(t)dt
if we restart the experiment n times on experiment I
Xi = X(33)
X = 1/n∑ xi
As n → ∞, X → E[X] (expected value)
In general x(t) ≠ X → E[X], x(t) is an ergodic process when
they are equal.