1. EE255/CPS226 Discrete Random Variables
Dept. of Electrical & Computer engineering
Duke University
2/22/2019

2. Discrete Random Variables
A random event is not necessarily a number e.g
(T,F), (R,Y,B), (american, british,..,zimbabwe) etc.
A random variable (rv) X is a mapping (function)
X is more accurately known as the IMAGE of s
Inverse image
Number : integer or real valued.
Examples of discrete RVs are, no. job or pkt arrivals in unit time, no. of rainy days in year etc.
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

3. Probability Mass Function (pmf)
Ax : set of all samples (events) such that,
pmf
Pmf may also be called discrete density function.
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

4. Pmf Properties If X is a finite or a countably infinite set values,
above property gets redefined as,
All rv values satisfy p1.
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

5. Distribution Function
pmf: defined for a specific rv value, i.e.,
Probability of a set
Cumulative Distrib Func (CDF)
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

6. Distribution Function (contd.)
For integer valued X,
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

7. Some common discrete distributions
Constant
Discrete Uniform
Bernoulli
Binomial
Geometric
Poisson
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

8. Constant Random Variable
1.0 c
1.0 c
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

9. Discrete Uniform Distribution
Discrete rv X that assumes n discrete value with equal probability 1/n
Discrete uniform pmf
Discrete uniform distribution function:
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

10. Bernoulli Distribution
Bernoulli distribution  RV generated by a single Bernoulli trial that has a binary valued outcome {0,1}
1.0
P+q=1 q x
0.0
1.0
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

11. Binomial Distribution
Binomial distribution  multiple Bernoulli trials (BTs)
RV Yn: no. of successes in n BTs, i.e. {1,0,0,1,0,1,1…}
Above is the equation for the Binomial pmf (p,n)
Binomial CDF
Notes on pk: pk term signifies n-successes.
(cnk : is caused by the fact that there are these many ways in which k 1’s may appear in n-long
sequence of 1’s and 0’s e.g. (0,0,1,0,1,1,1,0,0,1,0,0,… 1)
Important: each trial is assumed to be an independent trial.
Example 2.3 notes: 3-Bernoulli trials has 8-possible outcomes,
{000, 001, 010, 100, 011, 101, 110, 111}
FX (0)  0 successes: Prob. Of event (000) = 0/125
FX (1)  at least 1 success (i.e. 0 or 1 successes): Prob. Of event ( ) = 4x0.125=0.5
FX (2)  at least 2 successes: Prob. of events (000)= FX (1) + Prob( ) =1-Prob(111)=0.75
FX (3)  1
Symmetric, +ve skewed and –ve skewed Binomial: p=0.5, < 0.5 and > 0.5
As no. of trials n increases (to infinity), B(k;n,p) can be approx. to a normal (Gaussian) distribution
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

12. Geometric Distribution
Multiple Bernoulli trials  occurrence of 1st success.
In general, S may have countably infinite size
Z has image {1,2,3,….}. Assuming independent trials,
1. That is, count the no of trials until the 1st success occurs. Typical output is {00001…}
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

13. Geometric Distribution (contd.)
Geometric distribution is the only discrete distribution that exhibits MEMORY-LESS property.
Future outcomes are independent of the past events.
n trials completed with all failures. Y additional trials are performed before success, i.e. Z = n+Y or Y=Z-n
The last equation says that, conditioned on Z > n, the no. of trials remaining until 1st success i.e. Y=Z-n has the same pmf as Z had originally. In other words, the system does not remember how many failures it has already encountered.
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

14. Geometric Distribution (contd.)
Z rv: total no. of trials to 1st success. This count includes the successful trial.
Modified geometric pmf: does not includes the successful trial, i.e. Z=X+1. Then X follows modified geometric distrbution.
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

15. Nagative Binomial Distributiond.)
RV Tr: no. of trials until rth success.
Image of Tr = {r, r+1, r+2, …}. Define events:
A: Tr = n
B: Exactly r-1 successes in n-1 trials.
C: The nth trial is a success.
Clearly, since B and C are mutually independent,
We can now also define the modified –ve binomial distribution.
The resulting rv is defined as: Just the no. of failures until rth success.
Event A gets re-defined as, “Tr = n+r”  Event that there are n failures.
Event B: exactly r-1 successes in n+r-1 trials.
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

16. Poisson Distribution RV such as “no. of arrivals in an interval [0,t)”
In a small interval, Δt, prob. of new arrival= λΔt.
pmf b(k;n, λt/n); CDF B(k;n, λt/n)=
What happens when
The problem now become similar to Bernoulli trials and Binomial distribution.
Divide the interval [0,t) into n sub-intervals, each of length t/n. For a sufficiently large n,
These n intervals can be thought as constituting a sequence of Bernoulli trials, with success
probability p= λt/n .
So now the problem can again re-defined as, finding the prob of k arrivals in a total of n intervals each of duration t/n.
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

17. Poisson Distribution (contd.)
Poisson distribution often occurs in situations, such as, “no. of packets (or calls) arriving in t sec.” or “no. of components failing in t sec” etc.
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

18. Probability Generating Function (PGF)
Helps in dealing with operations (e.g. +, x) on rv’s
Letting, P(X=k)=pk , PGF of X is defined by,
One-to-one mapping: pmf (or CDF) PGF
See page 98 for PGF of some common pmfs
GX(z) is identical to the z-transform (digital filtering) of a discrete time function. If |z| < 1 (i.e. inside a unit circle), the above summation is guaranteed to converge to 1.
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

19. Discrete Random Vectors
Examples:
Z=X+Y, (X and Y are random execution times)
Z = min(X, Y) or Z = max(X1, X2,…,Xk)
X:(X1, X2,…,Xk) is a k-dim rv defined on S
For each sample point s in S,
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

20. Discrete Random Vectors (properties)
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

21. Independent Discrete RVs
X and Y are independent iff the joint pmf satisfies:
Mutual independence also implies:
Pair wise independence vs. set-wide independence
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

22. Discrete Convolution Let Z=X+Y . Then, if X and Y are independent,
In general, then,
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University