1. Random variables (r.v.) Random variable
Definition: a variable that takes on
various values in a random way
Examples
Number of items to check out at super-market
Discrete (takes on a countable number of values)
End-to-end delay in a communication network
Continuous ([propagation delay; +infinity])
Described thru
Probability mass function (pmf)
Probability density function (pdf)

2. Probability distribution functions
The behavior of a
Discrete random variable
is captured by a probability mass function
Example: # items to check out at super-market (X)
P[X = i] and i = 1, 2, . . ., 200 (a certain upper bound)
of a continuous random variable
is captured by a probability density function
Example: End-to-end delay in a communication network (D)
fD(d)

3. Probability theory Origin Experiments Example 1 Outcome Toss a coin
3 times
Possible r.v.
X = # of heads
A discrete r.v.
Outcome
H H H
H H T
H T H
H T T
T H H
T H T
T T H
T T T H T
Sample
space
1/2 H T H T H T

4. Example 1 continued What is the probability distribution of X?
Sample
space X
{0,1,2,3}
What is the probability distribution of X?
Building on the following observation
P(X = i) = # of feasible sample points / total # of sample points
P(X=0) = 1/8; P(X=1) = 3/8
P(X=2) = 3/8; P(X=3) = 1/8

5. Example 2: cost functions
X = cost_function (sample points)
Example
If heads appears you make 10 $
If tail appears you pay 5 $
X = amount of money you make
=> X = {-15, 0, 15, 30} (discrete r.v.)

6. Example 3: continuous r.v.
End to end delay in a communication network X
Continuous random variable
belongs to a state space
Lower bound = propagation delay
Upper bound = +infinity

7. Discrete random variables: probability mass functions
Classified according to their probability mass function
I will cover
Binomial distribution
Geometric distribution
Poisson distribution

8. Binomial distribution
is primarily associated with the tossing of a coin
A certain number of independent trials
Outcome #1 (or 1) with probability p (referred to as success)
Outcome#2 (or 2) with probability 1-p (referred to as failure)
Example: n trials (n=6)
What is the probability that the following sequence arises?
1, 2, 1, 2, 1, 2
Answer: Prob = p (1-p) p (1-p) p (1-p) = p3(1-p)3

9. Binomial random variable
Suppose
n independent trials resulting
in a “success” with probability p
And in a “failure” with probability (1-p)
If X represents the number of successes in the n trials
=> X is a binomial random variable with parameters (n, p)

10. Binomial distribution: example1
Four fair coins are flipped. What is the probability that two heads and two tails are obtained?
Solution
Let X equal the number of heads (successes)
=> X is a binomial r.v. with parameter (n=4, p=1/2)

11. Binomial distribution: example2
It is known
that any item produced by a certain machine will be defective
with probability 0.1, independent of any other item.
What is the probability that in
A sample of three items
At most one will be defective?

12. Geometric random variable
Experiment
n trials
Each having probability p of being a success
Are performed until a success occurs
If X is the number of trials required until the first success
X is a geometric r.v. with parameter p
Its probability mass function is

13. Geometric r.v.: application
Time sharing
Jobs running on a computer
gets queued in order to use the CPU
A quantum of time is assigned to each process
X: Represents how many times a job cycles around
=> is a geometric r.v.
6 tosses of a coin
The first outcome is Heads
How many more heads do I need before I get a tail?

14. Poisson distribution Poisson distribution is
Associated with the observation of event occurrences
If N represents the number of events in T
=> N/T = average number of events /minute
interested in answering the following question
How many occurrences of this event take place per minute?
The way it has been done
Either 0 or 1 event occurrence per minute
T=15 min
Event#1
Time

15. Poisson distributed random variable
A Poisson random variable X
Characterizes the number of occurrences of an event
Typically an arrival => X = # arrivals per unit time
With parameter λ (average # of arrivals per unit time)
The value of λ (arrival rate)

16. Poisson distribution: example 1
If number of accidents occurring on a highway per day
is a Poisson r.v. with parameter λ = 3,
What is the probability that no accidents occur today?
Solution

17. Poisson distribution: example 2
Consider an experiment that
counts the number of α-particles emitted in
a one-second interval by one gram of radioactive material.
If we know that ,on average , 3.2 such α-particles are given off
what is a good approximation
to the probability that no more than 2 α-particles appear?

18. Binomial approximation to the Poisson distribution
N events T
Time ΔT
Divide the time axis into ΔT
small enough so that
At most only one arrival can occur
n ΔTs are required
It is like creating a binomial experiment
Each ΔT is a trial
Outcome: 0 arrivals (p ?) or 1 arrival ((1-p)?)

19. Binomial approximation to the Poisson distribution (cont’d)
With what probability we are going to have
1 arrival or 0 arrivals in one ΔT ?
Average arrival rate per ΔT interval
As such

20. Binomial approximation to the Poisson distribution (cont’d)
If you let n tends to infinity you will get

21. Cumulative distribution
Consider a discrete r.v. X
Taking on the values from 0 to infinity
The cumulative distribution function can be expressed
F (j) = P(X <= j) = P(X=0) P(X=j-1) + P(X=j)
For instance
Suppose X has a probability mass function given by
P(1) = ½, P(2) = 1/3, P(3) = 1/6
The cumulative function F of X is given by

22. Residual distribution
Given by

23. Expectation of a discrete random variable
X is a discrete random variable
Having a probability mass function p(X)
=> Expected value of X is defined by E[X] as
Variance of X
a and b are constants

24. Expectation: example 1 Find E[X] Find Var(X)
where X is the outcome when we roll a fair dice
Find Var(X)
when X represent the outcome when we roll a fair dice

25. Expectation: example 2 Calculate E[X] when X is Binomially distributed
With parameters n and p

26. Expectation: example 3 Find E[X]
Of a geometric random variable X with parameter p

27. Expectation: example 4 Calculate E[X]
For Poisson random variable X with parameter λ