1. Engineering Probability and Statistics - SE-205 -Chap 3By
S. O. Duffuaa

2. Lecture Objectives Present the following: Concept of random variables
Probability distributions
Probability mass function

3. Random Experiment and random variables
Throwing a coin
S = { H, T}.
Define a mapping X: { H, T}  R
X(H) = 1 and X(T) = 0, Also the probability of 1 and 0 are the same as for H and T.
Then we call X a random variable.

4. Random Experiment and random variables
In the experiment on the number of defective parts in three parts the sample space S = { 0, 1, 2, 3}
Find P(0), P(1), P(2) and P(3)
P(0) = 1/8, P(1) = 3/8, P(2) = 3/8
and P(3) = 1/8

5. Probability Mass FunctionX o 1 2 3
f(x)
1/8
3/8
3/8
1/8
Properties of f(x)
f(x)  0
 f(x) = 1
Give many examples in class

6. Probability Mass Function
Build the probability mass functions for the following random variables:
Number of traffic accidents per month on campus.
Class grade distribution
Number of “ F” in SE 205 class per semester
Number of students that register for SE 205 every semester.

7. Cumulative Distribution Function
It is a function that provide the cumulative probability up to a point for a random variable (r.v). Defined as follows for a discrete r.v:
P( X  x) = F(x) =  f(t)
t x

8. Cumulative Distribution Function (CDF)
Example of a cumulative distribution function
F(x) = x  -2
=  x  0
=  x  2
=  x
What is the mass function for the above F(x). Note you need to subtract

9. Probability Mass Function Corresponding to Previous CDFX -2 2
f(x)
0.2
0.5
0.3
The above mass function is the one corresponding to the previous CDF is

10. Mean /Expected Value of a Discrete Random Variable (r.v)
The mean of a discrete r.v denoted as E(X) also called the expected value is given as:
E(X) = μ =  x f(x)
The expected value provides a good idea a bout the center of the r.v.
compute the mean of the r.v in previous slide:
E(X) = (-2) (0.2) + (0) (0.5) + (2)(0.3) = 0.2 x

11. Variance of A Random Variable
The variance is a measure of variability.
What is variability?
The variance is defined as:
V(X) =σ2 = E(X-μ)2 =  (x-μ)2f(x)
Compute the variance of the r.v in the slide before the previous one.
σ2 = (-2-0.2)2 (0.2) + (0-0.2)2(0.5) + (2-0.2)2(0.3) =
Also see example 3-9 and 3-11in the text.

12. Expected Value of a Function of a r.v
Let X be a r.v with p.m.f f(x) and let h(X) is a function of X. Then the expected value of h(X) is given as:
E(H(X)) =  h(x) f(x)
Compute the expected value of h(X) = X2 - X for the r.v in the previous slides.
See example 3-12 in text book. x

13. Discrete Random Variables
In this section will study several discrete distributions. For each distribution the student must be familiar with the following about each distribution:
Range and probability mass function
Cumulative distribution function
Mean and variance
2-3 applications

14. Discrete Random Variables
The following distributions will be studied
Discrete uniform
Bernoulli
Binomial
Hyper-geometric
Geometric
Poisson

15. Discrete Uniform
A random variable is discrete uniform if every point in its range has the same probability. If there are n points in the range, then the probability of each point is
f(x) = 1/n
An alternative way of defining uniform as follows: Suppose the rang is a, a+1, a+2, … b
The number of points is (b-a+1)
f(x) = 1/(b-a+1) for x = a, a+1, a+2, …, b

16. Discrete Uniform
The CDF F(x) you just multiply by the number of points less than or equal to x
The mean of the uniform is (b+a)/2
The variance of it is [(b-a+1)2 – 1]/12
Applied to following situations:
Random number generation
Drawing a random sample
Situation where vales have equal probabilities.

17. Bernoulli Trials
A trial with only two possible outcomes is used so frequently as a building block of a random experiment that it is called a Bernoulli trial. It is usually assumed that the trials that constitute the random experiment are independent. This implies that the outcome from one trial has no effect on the outcome to be obtained from any other trial. Furthermore, it is often reasonable to assume that the probability of a success in each trial is constant.

18. Bernoulli Trials
If we denote success by 1 an failure by 0, then the probability mass function f(x) is given as:
f(1) = p and f(0) = 1-p = q, as you see the range is 0 and 1
F(x) simple
Mean =E(X) = p
Variance = σ2 = p(1-p) = pq
Applications:
Building block for other distributions
Experiment with two outcomes

19. Binomial Random Variable
A random experiment consisting of n repeated trails such that
the trials are independent,
each trial results in only two possible outcomes, labeled as “success” and “failure”, and
the probability of a success in each trial, denoted as p, remeins constant
is called a binomial experiment.
The random variable X that equals the number of trials that result in a
success has binomial distribution with parameters p and n = 1, 2, ….
The probability mass function of X is

20. Binomial Random Variable
0.18
0.15
0.12
0.09
0.06
0.03
f(x) x x
0.4
0.3
0.2
0.1
f(x)
n p
n p
Figure 4-6 Binomial distributions for selected values of n and p

21. Binomial Random Variable
If X is a binomial random variable with parameters p and n, then
and
Applications:
Design of sampling plans for quality control
Estimation of product defects.

22. Geometric Random Variable
In a series of independent Bernoulli trials, with constant probability p of a success, let the random variable X denote the number of trials until the first success. Then X has a geometric distribution with parameters p and

23. Geometric Distribution
If X is a geometric random variable with parameters p, then the mean and variance
and
Applications:
Quality control, design of control charts
Estimation

24. Hyper-Geometric Distribution
A set of N objects contains
K objects classified as successes and
N – K objects classified as failures
A sample of size of n objects is selected, at random (without replacement)
from the N objects, where K  N and n  N
Let the random variable X denote the number of successes in the sample.
Then X has a hypergeometric distribution and

25. Hyper-Geometric Distribution
If X is a hypergeometric random variable with parameters N, K and n, then the mean and variance of X are
and
where p = K/N
Applications:
Design of inspection plans for quality control
Design of control charts

26. Poisson Random Variable
Given an interval of real numbers, assume counts occur at random throughout
the interval. If the interval can be partitioned into subintervals of small enough
length such that
the probability of more than one count in a subintervals is zero,
the probability of one count in a subinterval is the same for all
subintervals and proportional to the length of the subinterval, and
the count in each subinterval is independent of other subintervals,
then the random experiment is called a Poisson process
If the mean number of counts in the interval is  > 0, the random variable X
that equals the number of counts in the interval has a Poisson distribution
with parameters , and the probability mass function of X is

27. Poisson Random Variable
If X is a Poisson random variable with parameters , then the mean and variance of X are
and
Applications:
Model number of arrivals to a service facility
Model number of accidents per month
Demand for spare parts per month