ii. characterization of random variables. © tallal elshabrawy 2 random variable characterizes a...
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II. Characterization of Random Variables
© Tallal Elshabrawy 2
Random Variable
Characterizes a random experiment in terms of real numbers
Discrete Random Variables The random variable can only take a finite number of
values
Continuous Random Variables The random variable can take a continuum of values
© Tallal Elshabrawy 3
Probability Mass Function
Only Suitable to characterize discrete random variables
P X x
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Cumulative Distribution Function
1 2 1 2
1 2 1 2 2 1
F 0 F 1 0 F 1
F is monotone non-decreasing function
For < F F
For < P =F F
X X X
X
X X
X X
x
x
x x x x
x x x X x x x
F P X x X x
Properties of FX x
© Tallal Elshabrawy 5
Probability Density Function
Used to characterize Continuous Random Variables
2
1
1 2 2 1
1 2 2 1
1 2
df F
df 0
F f d f d 1
P P P
P F F
P f d
X X
X
x
X X X
X X
x
X
x
x xx
x
x y y x x
x X x X x X x
x X x x x
x X x x x
© Tallal Elshabrawy 6
Uniform Random Variable
xf x
a b
a b
1
0
1f
0
0
F
1
X
X
x a
x a x bb a
x b
x a
x ax a x b
b ax b
fX x
FX x
1
b a
© Tallal Elshabrawy 7
Gaussian Random VariableMany physical phenomenon can be modeled as Gaussian Random Variables most popular to communication engineers is … AWGN Channels
2
22 2
2
1~N , f
2
x m
XX m x e
mean
standard deviation
2
2
2
2
2
2
1F d 1 Q
2where
1Q d
2
y mx
X
t
x
x mx e y
x e t
© Tallal Elshabrawy 8
Exponential Random Variable
Commonly encountered in the study of queuing systems
1f 0
x
bX x e x
b
F 1 0
x
bX x e x
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How to Characterize a Distribution
Client: Tell me how good is your network?
Salesman: Well, P(Delay<1)=0.1, P(Delay<2)=0.3,P(Delay<3)=0.2,……
Client: HmmmSo what does this really mean?
Salesman: How can I explain this?
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Mean of Random Variables
Client: Tell me how good is your network?
Salesman: Well, The average delay per packet is 1 sec
Client: HmmmSo what does this really mean?
Salesman: If you need to send 100 packets, they will most likely take 100 seconds
© Tallal Elshabrawy 11
Mean of a Random Variable
E P X x X x
E f d
XX x x x
Discrete Random Variable
Continuous Random Variable
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Consider a Network where the delay ‘D’ is either 1 or 5 secondsi.e., P[D = 1] = 0.3, P[D =5] = 0.7
P[D = 0, 2, 3, 4] = 0, P[D = 6, 7, 8, 9, …] = 0
What is the mean delay? Let assume 100 packets, then most likely
30 packets will be delayed for 1 sec
70 packets will be delayed for 5 sec Therefore 100 packets will most likely take 30x1+70x5 = 380 sec Average Delay = 380/100 = 3.8 sec E[D] = 1xP[D=1]+5xP[D=5] = 3.8 sec
Example
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Moments of a Random Variable
Discrete Random Variable
Continuous Random Variable
E P n nX x X x
E f d
n nXX x x x
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Central Moments
E E E P n n
X X x X X x
E E E f d
n n
XX X X X x x
Discrete Random Variable
Continuous Random Variable
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Variance
Variance is a measure of random variable’s randomness around its mean value
2
22
Var =E E
Var =E E
X X X
X X X
Std = VarX X
© Tallal Elshabrawy 16
Conditional CDF
Define FX|A[x] as the conditional cumulative distribution function of the random variable X conditioned on the occurrence of the event A, then
P ,F P
P
X A
X x Ax X x A
A
P ,P
P
A BB A
A
Remember Bayes’s Rule
1 2 1 2
1 2 1 2 2 1
F 0 F 1 0 F 1
F is monotone non-decreasing function
For < F F
For < P =F F
X A X A X A
X A
X A X A
X A X A
x
x
x x x x
x x x X x A x x
Properties of FX A x
© Tallal Elshabrawy 17
Conditional CDF: Example
Consider a uniformly distributed random variable X with CDF
0 0
F 0 1
1 1
X
x
x x x
x
Calculate the conditional CDF of X given that X<1/2. In other words we would like to compute FX|X<1/2[x]
x
FX[x]
0 1
1
1 2
P , 1 2F P 1 2
P 1 2
X X
X x Xx X x X
X
1 2
0 P , 1 2 0
F 0 0
X X
x X x X
x x
© Tallal Elshabrawy 18
Conditional CDF: Example
0 0
F 0 1
1 1
X
x
x x x
x
Calculate the conditional CDF of X given that X<1/2. In other words we would like to compute FX|X<1/2[x]
x
FX[x]
0 1
1
1 2
P , 1 2F P 1 2
P 1 2
X X
X x Xx X x X
X
1 2
0 1 2 P , 1 2 P
PF 2 0 1 2
P 1 2
X X
x X x X X x
X xx x x
X
Consider a uniformly distributed random variable X with CDF
© Tallal Elshabrawy 19
Conditional CDF: Example
0 0
F 0 1
1 1
X
x
x x x
x
Calculate the conditional CDF of X given that X<1/2. In other words we would like to compute FX|X<1/2[x]
x
FX[x]
0 1
1
1 2
P , 1 2F P 1 2
P 1 2
X X
X x Xx X x X
X
1 2
1 2 P , 1 2 P 1 2
P 1 2F 1 1 2
P 1 2
X X
x X x X X
Xx x
X
Consider a uniformly distributed random variable X with CDF
© Tallal Elshabrawy 20
Conditional CDF: Example
0 0
F 0 1
1 1
X
x
x x x
x
Calculate the conditional CDF of X given that X<1/2. In other words we would like to compute FX|X<1/2[x]
x
FX[x]
0 1
1
1 2
0 0
F 2 0 1 2
1 1 2
X X
x
x x x
x x
FX[x]
0 1/2
1
Consider a uniformly distributed random variable X with CDF
© Tallal Elshabrawy 21
Exercise
For some random variable X and given constants a, b such that a<b
0
F FF
F F
1
X XX a X b
X X
x a
x ax a x b
b a
x b
P ,F P
P
X a X b
X x a X bx X x a X b
a X b
© Tallal Elshabrawy 22
Conditional PDF
Define fX|A[x] as the conditional probability density function of the random variable X conditioned on the occurrence of the event A, then
df F
dX A X Ax x
x
2
1
1 2
f 0
F f d f d 1
P f d
X A
x
X A X A X A
x
X Ax
x
x y y x x
x X x A x x
Properties of fX A x
© Tallal Elshabrawy 23
Conditional PDF: Example
Consider a uniformly distributed random variable X with CDF
0 0
f 1 0 1
0 1
X
x
x x
x
Calculate the conditional PDF of X given that X<1/2. In other words we would like to compute fX|X<1/2[x]
x
fX[x]
0 1
1
x
fX|X<1/2[x]
1
2
1 2
0 0
f 2 0 1 2
0 1 2
X X
x
x x
x
1/2
© Tallal Elshabrawy 24
Exercise
For some random variable X and given constants a, b such that a<b
0
F FF
F F
1
X XX a X b
X X
x a
x ax a x b
b a
x b
df F
d X a X b X a X bx xx
0
f ff
F F P
0
X XX a X b
X X
x a
x xx a x b
b a a x b
x b
© Tallal Elshabrawy 25
Conditioning on a Characteristic of Experiment
Conditioning does not necessarily have to be on the numerical outcome of an experiment
It is possible to have qualitative conditioning based on a characteristic of an experiment
Example: Consider a random variable X that represents the score of students in a given course Conditioning based on experiment outcome
The distribution of grades given it is greater than 80% (i.e., FX|X>80[x])
Conditioning based on experiment characteristic The distribution of grades given the gender of students (i.e., FX|M[x])
© Tallal Elshabrawy 26
Conditioning on a Characteristic of Experiment
Consider a set of N mutually exclusive events A1, A2,…, AN. Suppose we know FX|An[x] for n=1, 2, …, N. Then
1
1
1
F P P P
F P F P
f f P
n
n
N
X n nn
N
X nX An
N
X nX An
x X x X x A A
x X x x A
x x A
The unconditional CDF/PDF is basically the conditioned CDF averaged across the probability of occurrence of conditioning events
Example:
For a bit b sent over a communication channel and the received voltage r
P[r<0]=P[r<0|b=1]*P[b=1]+P[r<0|b=0]*P[b=0]
© Tallal Elshabrawy 27
Conditioning on a Characteristic of Experiment
Consider a set of N mutually exclusive events A1, A2,…, AN. Suppose we know FX|An[x] for n=1, 2, …, N. Then
P PP
P
n n
n
X x A AA X x
X xDiscrete Random Variable
For a continuous random variable P [X=x|An]=0, P [X=x]=0 resulting in an undetermined expression
© Tallal Elshabrawy 28
Conditioning on a Characteristic of Experiment
Consider a set of N mutually exclusive events A1, A2,…, AN. Suppose we know FX|An[x] for n=1, 2, …, N. Then for a continuous random variable
P PP
P
0
P f P f
f P f PP
f f
0
f PP
f
n
n n
n
n nn
n XX A
n nX A X A
nX X
nX A
nX
x X x A AA x X x
x X x
x X x A x x X x x
x A x AA x X x
x x
x AA X x
x
© Tallal Elshabrawy 29
Conditional Expected Value
The expected value of a random variable X conditioned on some event A
E P X AX A x x Discrete Random Variable
E f d
X AX A x x x Continuous Random Variable