1. The document discusses probability concepts related to single and multivariate random variables including probability density functions, cumulative distribution functions, expected value, variance, and common distributions. 2. It also covers topics related to bivariate and multivariate random variables such as joint probability, marginal probability, conditional probability, and the bivariate normal distribution. 3. The document then discusses random signals and linear estimation methods, including maximum likelihood estimation and mean square error estimation. It provides an example of estimating a signal using mean square error minimization.

1. Stochastic Section # 5
Revision Section
Eslam Adel
March 21, 2018
1 Single Random Variable
1.1 Probability density function PDF
It is defined as
f(x) = P(X = x) (1)
Properties
x
f(x)dx = 1 (2)
or for discrete random variables
x
f(x) = 1 (3)
1.2 Cumulative distribution function (CDF)
CDF is defined as
F(x) = P(X < x) (4)
Properties
(i) CDF is always continuous for discrete and continous RV.
(ii) CDF is increasing function
(iii) Maximum Value of CDF is 1
Example :
Let F(x) =
cx x = 1, 2, 3, 4, 5
0 otherwise
So 5c = 1 and c = 1
5
1.3 Expected Value
E[X] =
x
Xf(x)dx (5)
Notes
1. E[X] is a linear operator → E[aX + b] = aE[X] + b
2. E[const] = const
1.4 Variance
V ar[X] = E[(X − m)2
] = E[X2
] − (E[X])2
= E[X2
] − m2
(6)
1

2. 1.5 Common Distributions
1.5.1 Normal(Gaussian) Distribution
1.5.2 Uniform Distribution
1.5.3 Exponential Distribution
1.5.4 Rayleigh Distribution
2 Bi-variate Random Variables
2.1 Joint Probability
∞
−∞
∞
−∞
f(x, y)dxdy = 1 (7)
2.2 Marginal probability
fx(x) =
∞
−∞
fx,y(x, y)dy (8)
fy(y) =
∞
−∞
fx,y(x, y)dx (9)
2.3 Conditional probability
fx|y(x|y) =
fx,y(x, y)
fy(y)
(10)
fy|x(y|x) =
fx,y(x, y)
fx(x)
(11)
2.4 Bi-Variate Normal Distribution
f(x, y) =
1
2πσxσy 1 + ρ2
e
−1
2(1−ρ2)
[ x2
σ2
x
−2ρ xy
σxσy
+ y2
σ2
y
]
(12)
ρ is the correlation coefficient
ρ =
Cov(x, y)
σxσy
=
E[(x − mx)(y − my)]
σxσy
(13)
3 Multi-variate Random Variable
Multi-variate random variable or random vector X has N random variables
X =





X1
X2
...
XN





Note:
Each random variable X1 → XN is a random variable (has multiple observations).
3.1 Mean Vector
For random vector X mean vector m = E[X]
E[X] = E





X1
X2
...
XN





=





E[X1]
E[X2]
...
E[XN ]





2

3. 3.2 Correlation Matrix
R = E[XXT
] = E





X2
1 X1X2 . . . X1XN
X1X2 X2
2 . . . X2XN
...
...
...
...
X1XN . . . . . . X2
N





(14)
3.3 Covariance Matrix
= E[(X − m)(X − m)T
] (15)
= E





(X1 − m1)2
(X1 − m1)(X2 − m2) . . . (X1 − m1)(XN − mN )
(X1 − m1)(X2 − m2) (X2 − m2)2
. . . . . .
...
...
...
...
(X1 − m1)(XN − mN ) . . . . . . (XN − mN )2





(16)
Relation Between and R
= R − mmT
(17)
3.4 Gaussian Random Vector
For a gaussian random vector X with dimension N the probability density function is
f(X) =
1
(2π)
N
2 (det( ))
1
2
e
−1
2 ((X−m)T −1
(X−m))
(18)
where det( ) is the determinant of covariance matrix,
−1
is the inverse of covariance matrix, and m is the
mean vector.
It is conventionally presented as X ∼ N(m, ) where m is mean vector and is the covariance matrix.
4 Random Signals
1. Random signal is a sequence of random variables [x1, x2, x3, . . . , xN ].
2. Mean of the random signal E[x] can be approximated as
E[x] =
1
N
N−1
i=0
xi (19)
3. Sample Autocorrelation Rxnxn+k
is defined as:
Rxnxn+k
=
1
N
N−1−k
i=0
xi+kxi k = 0, 1, 2, . . . N (20)
4. We always assume that the signal is zero mean. If the signal is not zero mean, subtract the mean from it.
x → x − E[x]
Note: for zero mean signal Rxx = σ2
5. For random signal it is assumed that it is a wide sense stationary signal (WSS) Where :
1. E[x] = constant
2. Rxn+kxn
= E[xn+kxn] = Rxx(k) (not a function of time variable n)
6. Stationarity means that for different windows of the same signal, statistical parameters are almost the
same.
7. For WSS signal Rxx(k) = Rxx(−k)
3

4. 5 Random Signal Models
5.1 Moving Average Model (MA)
X(Z) = C(Z) (Z)
Where (Z) is uncorrelated white gaussian noise (WGN)
In time domain
x(n) = N−1
i=0 cn−i i
5.2 Auto Regressive Model (AR)
X(Z) = b0
A(Z) (Z)
In time domain
N−1
i=0 an−ixi = b0 (n)
5.3 Auto Regressive Moving Average Model (ARMA)
X(Z) = B(Z)
A(Z) (Z)
In time domain
N−1
i=0 an−ixi =
N−1
i=0 bn−i i
6 Linear Estimation
6.1 Maximum Likelihood Method (ML)
Maximize the joint probability function of all random variables f(x1, x2, . . . , xN )
6.2 Mean Square Error Method (MS)
Error is defined as :
e = x − ˜x
So the mean square error will be
E[(x − ˜x)2
]
to minimize the error differentiation of MSe with respect to model parameters must be zero.
6.2.1 Lecture Example
˜x = hy + g (21)
Solution:
MSe = E[e2
] = E[(x − ˜x)2
] = E[(x − hy − g)2
]
∂MSe
∂h = 0
E[2(x − hy − g) × −y] = 0
−E[xy] + hE[y2
] + gE[y] = 0
hRyy(0) + gE[y] = Rxy(0) (22)
Similarly for g
4

5. ∂MSe
∂g = 0
E[2(x − hy − g) × −1] = 0
−E[x] + hE[y] + g = 0
hE[y] + g = E[x] (23)
We can put it in matrix form
1 E[y]
E[y] Ryy(0)
g
h
=
E[x]
Rxy(0)
(24)
Finally
g
h
=
1 my
my Ryy(0)
−1
mx
Rxy(0)
(25)
Ryy(0) = σ2
y + m2
y and Rxy(0) = ρσxσy + mxmy
Values of g and h are
g = mx(σ2
y + m2
y) − myρσxσy − mxm2
y
h = ρσx
σy
For Given Signals x = [−2, −1, 0, 1, 2] and y = [−5, −3, 0, 3, 5]
g = 0 and h = 0.38
we can get the estimate ˜x = [−1.91, −1.14, 0, 1.14, 1.91] and mean square error will be 0.011
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