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IOE/Stat 265, Fall 2009
IOE/Stat 265, Fall 2009
Lecture #6:
Lecture #6:
Discrete Distributions
Discrete Distributions
31
Discrete Random Variables
32
Probability Distributions for Discrete
Random Variables
3.3 Expected Values
3.4
Binomial Probability Distribution
3.5
Hypergeometric and Neg. Binomial
3.6
Poisson Probability Distribution
2
Ch 3:
Learning Objectives
1.
Determine probabilities from PMF and viceversa
2.
Determine probabilities from CDF and determine
CDF from PMF
3.
Calculate means and variances for discrete r.v.
4.
Learn some basic discrete PMFs and their
applications
5.
Calculate probabilities, means, and variances for
select discrete distributions.
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Recall: Random Variables
±
Definition: A random variable (r.v.) is a function
(rule) that assigns a real number to each
possible outcome in the sample space of a
random experiment.
±
Example:
if X is a random variable that
defines the number of heads that we may
obtain when we flip 3 coins, then X can take
on 4 possible values: 0, 1, 2, 3
±
A r.v. is denoted “X”.
The measured random
variable (aka realization) is denoted “x”
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Discrete or Continuous Variables
±
A discrete r.v. has finite range
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e.g. number of scratches in paint, proportion of
defects in 10 samples, number of errors.
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A continuous r.v. is defined on an interval of
real numbers.
±
e.g. pressure, temperature, voltage, current,
weight.
This chapter confines attention to
Discrete
case
∑
∫
versus
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Example:
Sum of Two Dice
±
Let X be a random variable that represents the sum of rolling
two dice. Fill in the blanks below with the values of X and its
frequency…
Combination
X
Frequency
(1,1)
(1,2) (2,1)
(1,3) (3,1) (2,2)
(1,4) (4,1) (2,3) (3,2)
(1,5) (5,1) (2,4) (4,2) (3,3)
(1,6) (6,1) (2,5) (5,2) (3,4) (4,3)
(2,6) (6,2) (3,5) (5,3) (4,4)
(3,6) (6,3) (4,5) (5,4)
(4,6) (6,4) (5,5)
(5,6) (6,5)
(6,6)
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3/36
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Example 2: Bernoulli Process
±
A special case of a discrete r.v. whose values
are only 0 or 1 is called a Bernoulli random
variable
±
Common Notation:
x = 0
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This note was uploaded on 03/17/2010 for the course IOE 265 taught by Professor Garyherrin during the Fall '09 term at University of MichiganDearborn.
 Fall '09
 GaryHerrin

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