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

© Tallal Elshabrawy 4 Cumulative Distribution Function

© Tallal Elshabrawy 5 Probability Density Function Used to characterize Continuous Random Variables

© Tallal Elshabrawy 6 Uniform Random Variable a b a b 1

© Tallal Elshabrawy 7 Gaussian Random Variable Many physical phenomenon can be modeled as Gaussian Random Variables most popular to communication engineers is … AWGN Channels mean standard deviation

© Tallal Elshabrawy 8 Exponential Random Variable Commonly encountered in the study of queuing systems

© Tallal Elshabrawy 9 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: Hmmm So what does this really mean? Salesman: How can I explain this?

© Tallal Elshabrawy 10 Mean of Random Variables Client: Tell me how good is your network? Salesman: Well, The average delay per packet is 1 sec Client: Hmmm So 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 Discrete Random Variable Continuous Random Variable

© Tallal Elshabrawy 12 Consider a Network where the delay ‘D’ is either 1 or 5 seconds i.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

© Tallal Elshabrawy 13 Moments of a Random Variable Discrete Random Variable Continuous Random Variable

© Tallal Elshabrawy 14 Central Moments Discrete Random Variable Continuous Random Variable

© Tallal Elshabrawy 15 Variance Variance is a measure of random variable’s randomness around its mean value

© Tallal Elshabrawy 16 Conditional CDF Define F X|A [x] as the conditional cumulative distribution function of the random variable X conditioned on the occurrence of the event A, then Remember Bayes’s Rule

© Tallal Elshabrawy 17 Conditional CDF: Example Consider a uniformly distributed random variable X with CDF Calculate the conditional CDF of X given that X<1/2. In other words we would like to compute F X|X<1/2 [x] x FX[x]FX[x] 0 1 1

© Tallal Elshabrawy 18 Conditional CDF: Example Calculate the conditional CDF of X given that X<1/2. In other words we would like to compute F X|X<1/2 [x] x FX[x]FX[x] Consider a uniformly distributed random variable X with CDF

© Tallal Elshabrawy 19 Conditional CDF: Example Calculate the conditional CDF of X given that X<1/2. In other words we would like to compute F X|X<1/2 [x] x FX[x]FX[x] Consider a uniformly distributed random variable X with CDF

© Tallal Elshabrawy 20 Conditional CDF: Example Calculate the conditional CDF of X given that X<1/2. In other words we would like to compute F X|X<1/2 [x] x FX[x]FX[x] x FX[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

© Tallal Elshabrawy 22 Conditional PDF Define f X|A [x] as the conditional probability density function of the random variable X conditioned on the occurrence of the event A, then

© Tallal Elshabrawy 23 Conditional PDF: Example Consider a uniformly distributed random variable X with CDF Calculate the conditional PDF of X given that X<1/2. In other words we would like to compute f X|X<1/2 [x] x fX[x]fX[x] x f X|X<1/2 [x] 1 2 1/2

© Tallal Elshabrawy 24 Exercise For some random variable X and given constants a, b such that a<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., F X|X>80 [x]) Conditioning based on experiment characteristic The distribution of grades given the gender of students (i.e., F X|M [x])

© Tallal Elshabrawy 26 Conditioning on a Characteristic of Experiment Consider a set of N mutually exclusive events A 1, A 2,…, A N. Suppose we know F X|An [x] for n=1, 2, …, N. Then 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 A 1, A 2,…, A N. Suppose we know F X|An [x] for n=1, 2, …, N. Then Discrete Random Variable For a continuous random variable P [X=x|A n ]= 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 A 1, A 2,…, A N. Suppose we know F X|An [x] for n=1, 2, …, N. Then for a continuous random variable

© Tallal Elshabrawy 29 Conditional Expected Value The expected value of a random variable X conditioned on some event A Discrete Random Variable Continuous Random Variable