Discrete Random Variables

1
Discrete Random Variables
2
Random Variables

• A random variable is a function that assigns
  numerical values to all the outcomes in the
  sample space.

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Notation

• We will use capital letters from the end of the
  alphabet to represent a random variable.
• Usually Y
• The corresponding lower case letter will
  represent a particular value of the random
  variable.
• P(Y y) is the probability that the random
  variable Y is equal to the value y.

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Discrete and Continuous Random Variables

• A continuous random variable can take any value
  in an interval of the real number line.
• Usually measurements
• A discrete random variable can take a countable
  number of distinct values.
• Usually counts

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Discrete and Continuous Random Variables

• Discrete or continuous?
• Time until a projectile returns to earth.
• The number of times a transistor in computer
  memory changes state in one operation.
• The volume of gasoline that is lost to
  evaporation during the filling of a gas tank.
• The outside diameter of a machined shaft.
• The number of chapters in your test book.

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Discrete Numeric Random Variables

• A small airport in New Zealand is interested in
  the number of late aircraft arrivals per day.
  Every day for a year it counts the daily number
  of late arrivals.
• Let random variable Y Number of aircraft in
  one day that arrive late.

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Discrete Numeric Random Variables

• What is the probability that on any randomly
  chosen day that 3 aircrafts are late?
• What is the chance that on any randomly chosen
  day at least 1 aircraft is late?

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Discrete Numeric Random Variables

• An assembly consists of three mechanical
  components. Suppose that the probabilities that
  the first, second, and third components meet
  their specifications are 0.90, 0.95 and 0.99
  respectively. Assume the components are
  independent.
• Let Si the event that component i is within its
  specification.

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Possible Outcomes for One assembly
0.00495
0.00005
Let Y Number of components within specification
in a randomly chosen assembly
Y 0 1 2
3 P(Yy) 0.00005 0.00635 0.14715
0.84645
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Discrete Random Variables

• Probability mass function
• p(y) P(Y y)
• Note
• 0 lt p(y) lt 1 for every value of y
• and

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Probability Function for a Discrete Random
Variable can be expressed as a table or graph and
sometimes as a formula.
Y 0 1 2
3 P(Yy) 0.00005 0.00635 0.14715
0.84645
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Cumulative Distribution Function

• If Y is a random variable, then the cumulative
  distribution function is denoted by F(y).
• F(y) P(Y lt y)

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Number of components within specification for a
randomly chosen assembly
Distribution Function
Y 0 1 2
3 P(Yy) 0.00005 0.00635 0.14715
0.84645
Cumulative Distribution Function
Y 0 1 2
3 P(Ylty) 0.00005 0.00640 0.15355
1.0000
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Cumulative Distribution Function
Distribution Function
P(Y lt y)
P(Y y)
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Expected Value

• Note, this distribution is the distribution for a
  population
• The expected value of the population is the
  population mean, µ, for the distribution.

Y 0 1 2
3 P(Yy) 0.00005 0.00635 0.14715
0.84645
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Number of Components within Specification on a
Randomly Chosen Assembly
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Variance

• The variance of the population is the expected
  value of (Y - µ)2.

Y 0 1 2
3 P(Yy) 0.00005 0.00635 0.14715
0.84645
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Computational Formula for Variance
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Population Standard Deviation
20
We receive 75 of a certain part from Supplier A
and 25 from supplier B. Two of these parts are
used in an assembly. Let Y number of parts from
Supplier A in the assembly. What is the expected
value of Y?
21
A project manager for an engineering firm
submitted bids on three projects. The following
table summarizes the firms chances of having the
three projects accepted.
Project A B C Prob
of accept 0.30 0.80 0.10
Assuming the projects are independent of one
another, what is the probability that the firm
will have all three projects accepted?
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What is the probability of having at least one
project accepted?
Project A B C Prob
of accept 0.30 0.80 0.10
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Let Y number of projects accepted.
y 0 1 2 3
p(y) 0.126 0.572 0.278 0.024
What is the expected number of projects accepted?
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Let Y number of projects accepted.
y 0 1 2 3
p(y) 0.126 0.572 0.278 0.024
What is the variance for number of projects
accepted?
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Rules of Expectation

• E(c) c
• E(cY) cE(Y)
• E(X Y) E(X) E(Y)

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Rules of Variance

• Var(c) 0
• Var(cY) c2Var(Y)
• Var(X Y) Var(X) Var(Y)
• when X and Y are independent.

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Binomial Random Variable

• Suppose we toss a fair coin 20 times, counting
  the number of heads.
• This is an example of a binomial experiment.
• The possible outcomes can be represented with a
  binomial random variable which follows a binomial
  distribution.

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Binomial Experiment

• There are n identical trials.
• There are two possible outcomes for each trial
  success and failure.
• Outcomes are independent from trial to trial.
• Probability of success, p, remains constant from
  trial to trial.
• Let Y denote the number of success. Then Y has a
  Binomial distribution, denoted by YB(n, p).

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Binomial Experiment

• Each toss of the coin is a trial. We performed 20
  trials. So n 20.
• The two possible outcomes of a trial are heads or
  tails. Since we are counting heads, heads is a
  success, while tails is a failure.
• Are the outcomes of our trials independent of one
  another?
• Probability of success (heads) is 0.50

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Binomial Experiment

• A manufacturer of water filters for refrigerators
  monitors the process for defective filters.
  Historically, this process averages 5 defective
  filters.
• Suppose five filters are randomly selected for
  testing. We are interested in the number of
  defectives in the sample.
• Define a trial.
• Calculate n.
• Define a success.
• Are the outcomes independent from trial to trial?
• Calculate p.

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Water Filters

• A manufacturer of water filters for refrigerators
  monitors the process for defective filters.
  Historically, this process averages 5 defective
  filters. Five filters are randomly selected.
• Find the probability that all five filters are
  defective.
• Find the probability that no filters are
  defective.
• Find the probability that exactly 1 filter is
  defective.

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• A manufacturer of water filters for refrigerators
  monitors the process for defective filters.
  Historically, this process averages 5 defective
  filters. Five filters are randomly selected.
• Find the probability that exactly 2 filters are
  defective.

• 5(.05)2(.95)3
• 10(.05)2(.95)3
• 15(.05)2(.95)3

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Combinations

• Number of ways to choose r distinct objects from
  n distinct objects

n choose r
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Recall

• n!
• 5! (5)(4)(3)(2)(1) 120
• 7! (7)(6)(5)(4)(3)(2)(1) 5040
• 1! 1
• 0! 1

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Probability Function for a Binomial Random
Variable
for y 0, 1, 2, ,n
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If Y follows a Binomial Distribution

• Binomial mean

• Binomial variance

• Binomial standard deviation

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Suppose we are producing 15 defective filters.
Let Y Number of Defective Filters in a Sample
of 10
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What is the approximate probability that there
will be at least 1 defective filter in a sample
of 10?
A. 0.20 B. 0.35 C. 0.80 D. 0.90
39
Historically, 10 of homes in Florida have radon
levels higher than that recommended by the EPA.
In a random sample of 20 homes, find the
probability that exactly 3 have radon levels
higher than the EPA recommendation.
40
If a manufacturing process has a 0.03 defective
rate, what is the probability that at least one
of the next 25 units inspected will be defective?

• (0.03)1 (0.97)24
• 1 (0.03)1 (0.97)24
• 1 (0.03)0 (0.97)25
• 1 (0.03)25

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A manufacturing process has a 0.03 defective
rate. If we randomly sample 25 units

• What is the probability that less than 6 will be
  defective?
• What is the probability that 4 or more are
  defective?
• What is the probability that between 2 and 5,
  inclusive, are defective?

42
Insulated Wire

• Consider a process that produces insulated copper
  wire. Historically the process has averaged 2.6
  breaks in the insulation per 1000 meters of wire.
  We want to find the probability that 1000 meters
  of wire will have 1 or fewer breaks in
  insulation?
• Is this a binomial problem?

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Poisson Distribution

• Poisson distribution can be used to model the
  number of events occurring in a continuous time
  or space.
• Let Y number of breaks in 1000 meters of wire.
• P(Y lt 1) P(Y 0) P(Y 1)

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Poisson Distribution
for y 0, 1, 2,
where ? is the average number of occurrences per
base unit. and t is the number of base units
inspected.
Further
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Insulated Wire
Let Y Number of breaks in 1000 meters of wire.
? 2.6 and t 1
The expected number of breaks in 1000 meters of
wire is 2.6.
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Insulated Wire

• If we were inspecting 2000 meters of wire, ?t
  2.62 5.2
• If we were inspecting 500 meters of wire, ?t
  2.60.5 1.3

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Conditions for a Poisson Distribution

• Areas of inspection are independent of one
  another.
• The probability of the event occurring at any
  particular point in space or time is negligible.
• The mean remains constant over all areas of
  inspection.

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Suppose we average 5 radioactive particles
passing a counter in 1 millisecond. What is the
probability that exactly 3 particles will pass in
one millisecond?
49
Suppose we average 5 radioactive particles
passing a counter in 1 millisecond. What is the
probability that exactly 10 particles will pass
in the next three milliseconds?