Engineering Probability and Statistics - SE-205 -Chap 3

1
Engineering Probability and Statistics - SE-205
-Chap 3

• By
• S. O. Duffuaa

2
Lecture Objectives

• Present the following
• Concept of random variables
• Probability distributions
• Probability mass function

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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.

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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

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Probability Mass Function

X
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
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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.

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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
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Cumulative Distribution Function (CDF)

• Example of a cumulative distribution function
• F(x) 0 x ? -2
• 0.2 -2 ? x ? 0
• 0.7 0 ? x ? 2
• 1.0 2 ? x
• What is the density function for the above
  F(x). Note you need to subtract

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Probability Mass Function Corresponding to
Previous CDF

X
-2
0
2
f(x)
0.3
0.5
0.2
The above density function is the one
corresponding to the previous CDF is
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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
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Variance of A Random Variable

• The variance is a measure of variability.
• What is variability?
• The variance is defined as
• V(X) s2 E(X-µ)2 ? (x-µ)2f(x)
• Compute the variance of the r.v in the slide
  before the previous one.
• s2 (-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.

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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
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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

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Discrete Random Variables

• The following distributions will be studied
• Discrete uniform
• Bernoulli
• Binomial
• Hyper-geometric
• Geometric
• Poisson

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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, a1, a2, b
• The number of points is (b-a1)
• f(x) 1/(b-a1) for x a, a1, a2,
  , b

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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 (ba)/2
• The variance of it is (b-a1)2 1/12
• Applied to following situations
• Random number generation
• Drawing a random sample
• Situation where vales have equal probabilities.

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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.

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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 s2 p(1-p) pq
• Applications
• Building block for other distributions
• Experiment with two outcomes

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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
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Binomial Random Variable

• n p
• 20 0.5

• n p
• 10 0.1
• 10 0.9

Figure 4-6 Binomial distributions for selected
values of n and p
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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.

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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
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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

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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
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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

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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 ?
  gt 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

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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