Chapter Three Discrete Random Variables

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Chapter ThreeDiscrete Random Variables
Probability Distributions
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Random VariableA rule that assigns a number to
each outcome in the sample space.
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Types of RVsDiscrete Possible values either
constitute a finite set or else an infinite
sequence in which there is a first element,
second element, etc.Continuous Possible values
consists of an entire interval on the number line.
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Bernoulli RVA RV with only two possible values
0 or 1
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Probability DistributionA mathematical model
that relates the value of a RV with the
probability of occurrence of that value in the
population.
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Probability MeasuresGiven real numbers r1
r2Probability that a RVa) Equals r1 b) Is
greater than r1 c) Is between r1 r2 d) Is
less than r1e) Is less than or equal to r1
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Example Discrete Probability DistributionA
machine produces 3 items per day. QC inspection
assigns to each item at the end of the day
defective or non-defective. Assume that each
point in the sample space has equal probability.
If RV X is the number of defective units at the
end of the day, what is the probability
distribution for X?
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Example Discrete Probability DistributionContinu
ed from Previous ExampleIf a non-defective
items yields a profit of 1,000 whereas a
defective item results in a loss of 250. What is
the probability distribution for the total profit
for a day?
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Discrete Probability DistributionA mfg. plant
has 3 student 3 veteran engineers assigned to
the shop floor. Two engineers are chosen at
random for a special project. Let the RV X denote
the number of student engineers selected. Find
the probability distribution for X.
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Discrete Probability DistributionStarting at a
fixed time, we observe the make of each car
passing by a certain point until a Ford passes
by. Let p P(Ford) RV X defined as the number
of cars observed. Find the probability
distribution for X.
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Example PMFConsider the number of cells exposed
to antigen-carrying lymphocytes in the presence
of polyethylene glycol to obtain first fusion.
The probability that a given cell will fuse is
known to be ½. Assuming that the cells behave
independently, find the probability distribution
for the number of cells required for first
fusion. What is the probability that four or
more cells require exposure to obtain the first
fusion?
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Cumulative Distribution FunctionCDF of a
Discrete RV X with pmf p(x) is defined for every
number x byF(x) P(Xltx) ? p(y)
y yltxORF(n) p(x1) p(x2)
p(xn)Where xn is the largest value of the xs
less than or equal to n.
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CDF PropertiesFor any two numbers a b with a
lt b, P(altXltb) F(b) F(a-)Where a-
represents the largest possible X value that is
strictly less than a.If all integers for a, b,
and values P(alt Xltb) F(b)
F(a-1)Taking ab P(a) F(a) F(a-1)
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Example CDFLet X denote the RV which is the toss
of a loaded die. The probability distribution of
X followsp(1) p(2) 1/6 p(3) 1/12p(4)
p(5) ¼ p(6) xa) Find the value of x.b)
Evaluate the CDF at 3.6.c) Find p(3ltXlt5).
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CDF to Find ProbabilitiesA mail-order business
has 6 telephones. Let RV X denote the number of
phones in use at a specified time. The pmf of X
is given as follows x 0 1 2 3 4 5 6p(x)
.01 .03 .13 .25 .39 .17 .02What is
the probability of a) at most 3 lines are in
use?b) at least 5 lines are in use?c) between 2
4 lines, inclusive are in use?
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Expected Values Of Discrete RV E(X) ?X ?
xnp(xn) All n Multiply every value
that the RV can take on by the probability that
it takes on this value then add all of these
terms together.
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Example of Expect ValuesWhat is the expected
value of the RV X where X is the value on the
face of a die?
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Expected ValueWhat is the expected value for
the RV X, which is the sum of the upturned faces
when two dice are tossed?
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Expected ValueA university has 15,000 students.
Let RV X equal the number of courses for which a
randomly selected student is registered. The pmf
of X follows x 1 2 3 4 5
6 7 p(x) .01 .03 .13 .25 .39 .17
.02Find the expected value of X.
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Expected Value of Bernoulli RV pmf p(x) 1
p for x 0 p
for x 1What is the expected value of X?
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Expected Value of a Function Eh(X) ?
h(k)p(k) All
kRV X has set of possible values k pmf p(x).
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Variance of a Discrete RV V(X) ?2 ? (k -
?)2p(k) All kRV X has a set of
possible values k with pmf p(x) and expected
value ?. SD(X) ?X v(?X2)
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Variance Shortcut MethodV(X)? k2p(k) - ?2
E(X2)-E(X)2 All k Steps
Find E(X2) Compute E(X) Square E(X) Subtract
this value from E(X2)
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Example of Variance E(X)A discrete pmf is
given by p(x) Ax x 0,1,2,3,4,5
Determine A. What is the probability that xlt3?
What is the expected value of X? What is the
Variance SD?
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Example of E(X) VarianceGiven the following
pmf for RV X x p(x) 0 1/8 1 ¼ 2 3/8 3 ¼Find
the E(X).Find the Variance. (Use Short Cut)
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Expect Value VarianceThree engineering
students volunteer for a taste test to compare
Coke Pepsi. Each student samples 2 identical
looking cups decides which beverage he or she
prefers. How many students do we expect to pick
Pepsi knowing that 3/5 of all students prefer
Pepsi over Coke?b) Find the Variance of the RV.
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Discrete Probability Distributions Binomial Nega
tive Binomial Hypergeometric Poisson
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Binomial Probability DistributionBinomial
Experimentgt Consists of a sequence of n
trials, where n is fixed in advance.gt The
trials are identical each trial can result
in one of the same two possible outcomes.gt
The trials are independent.gt The probability of
the outcomes is constant is equal to p
1-p.
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Binomial pmfb(x n, p) n! px
(1-p)n-x x!(n-x)! x 0,1,2,3,,n
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Example Discrete pmfYou draw at random a 20
piece sample from a group of 300 parts in
storage where 10 of the parts are known to be
out of specification. What is the probability
that 1 part in your sample will be out of spec.?
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Binomial ExampleA coin is tossed 4 times. What
is the pmf for the RV X the number heads?What
is the probability of having 3 or fewer Heads?
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Binomial ExampleThere are 5 intermittent loads
connected to a power supply. Each load demands
either 2w or no power. The probability of
demanding 2w is ¼ for each load. The demands are
independent. What is the pmf for the RV X, the
power required?
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Example Binomial pmfA lot of 300 manufactured
baseballs contains 5 defects. If a sample of 5
baseballs is tested, what is the probability of
discovering at least one defect.
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Negative Binomial pmfExperimentgt The trials
are independent.gt Each trial can result in
either a success (S) or a failure (F).gt The
probability of the outcomes is constant from
trial to trial.gt The experiment continues until
a total of r successes have been
observed, where r is a specified positive
integer of interest.
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Negative Binomial pmf nb(x r, p) (xr-1)!
pr (1-p)x (r-1)!(x)! x 0,1,2,3,
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Negative Binomial ExampleAn engineering manager
needs to recruit 5 graduating student engineers.
Let p P (a randomly selected student agrees to
be hired). If p 0.20, what is the probability
that 15 student engineers must be given an offer
before 5 are found who accept?
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Discrete pmf ExampleYour oil exploration crew is
testing for well sites. Historically, the
probability of finding oil in your present
geographical location is 1/20. HQs needs your
crew to locate 2 oil producing wells within 2
weeks. If set-up testing for oil takes 1 day,
what is the probability that it will take less
than 2 working weeks to find these 2 spots?
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nb Probability DistributionYou have passed your
ISE 261 at the end of the semester decide to
celebrate. For whatever reason, you are arrested
for a misdemeanor sentenced for 90 days in the
county jail. The judge being a student of
Probability Theory decides to give you an option.
You can have the full 90 days or you can elect to
leave jail after rolling 1 die for 16 straight
even numbers.Which option do you decide to take?
Remember, the judge will only give you a guard
for affirming your rolls for 8 hours per day.
Your ability to roll one die for 16 rolls is 2
minutes the judge insists on blocks of 16 rolls.
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Geometric Probability DistributionYou have
conducted a series of experiments to reduce the
proportion of scrapped battery cells to 1 in
your manufacturing plant. Now, what is the
probability of testing 51 cells without finding a
defect until the last cell?
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Hypergeometric pmfExperimentgt Consists of N
individuals, objects, or elements (a finite
population).gt Each individual can be
characterized as a success (S) or failure (F),
there a M successes in the population.gt A
sample of n individuals is selected without
replacement in such a way that each subset of
size n is equally likely to be chosen.
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Hypergeometric Probability DistributionFor RV X
the number of Ss in the sample. h(x n, M,
N) Cx,M Cn-x,N-M Cn,N
Integer x satisfies Max(0, n-NM)lt x lt
Min(n,M)
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Example Hypergeometric pmfFrom a group of 20 EE
students, you select 10 for employment. What is
the probability that the 10 selected include all
the 5 best engineers in the group of 20?
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Hypergeometric ExampleYour manufactured product
is shipped in lots of 20. Testing is costly, so
you sample production rather than use 100
inspection. A sampling plan constructed to
minimize the number of defectives shipped to
customers calls for sampling 5 items from each
lot rejecting the lot if more than 1 defective
is observed. If a lot contains 4 defectives, what
is the probability that it will be rejected?
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Poisson Probability DistributionFor RV X, the
number of random events that occur in a unit of
time, space, or any other dimension often
follows p(x ?) e-?(?)x x 0,1,2,
x! ? gt 0
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Binomial ApproximationIn any binomial experiment
in which n is large p is small, the binomial
pmf is approximately equal to the poisson pmf
where ? np.Rule of Thumbn gt 100 p lt .01
np lt 20
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Mean Variance of PoissonE(X) V(X) ?
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Poisson pmf ExampleYou are in charge of a PCB
operation. It is known that the distribution of
the number of solder balls occurring on 1 board
in this process is Poisson with ? 1.0. You have
1,000 PCBs in this process would like to
estimate how many boards would have solder balls
on them.
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Poisson Probability DistributionA radioactive
substance emits alpha particles. The number of
particles reaching a counter during an interval
of 1 second has been observed to have a Poisson
probability distribution with ? 10.What is
the probability that the RV X, the number of
particles reaching the counter during 1 second is
3.0?
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Poisson pmfA machine produces sheet metal where
the RV X, the number of flaws per yard follows a
Poisson distribution. The average number of flaws
per yard is 2.0. Plot the pmf for RVX.
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Poisson Approximation ExampleElectrical
resistors are packaged 200 to a continuous feed
ribbon. Historically, 1.5 percent of the
resistors manufactured by a machine are
defective. Compute the pmf of the number of
defective resistors on a ribbon and compare it to
the Poisson approximation. (Stop at x ?)
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Poisson Process Px(t) e-?t(?t)x
x! For time interval t with parameter
? ?t.
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Poisson Process ExampleAs systems engineering
manager, you have devised a random system of
police patrol so that an officer may only visit a
given location in his area with the Poisson RV X
0,1,2,3, times per 1-hour period. The system
is arranged so that he visits each location on an
average of once per hour. Calculate the
probability that an officer will miss a given
location during a half-hour period.What is the
probability that an officer will visit once?
Twice? At least once?
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Poisson pmfIn an industrial plant there is a dc
power supply in continuous use. The known failure
rate is ? 0.40 per year replacement supplies
are delivered at 6 months intervals. If the
probability of running out of replacement power
supplies is to be limited to 0.01, how many
replacement power supplies should the operations
engineer have on-hand at the beginning of the
6-month interval?
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Poisson ProcessThe number of hits on the ISE 261
Web site from Noon to 1230 PM on the day before
a Quiz follows a Poisson distribution. The mean
rate is 3 per minute. Find the probability that
there will be exactly 10 hits in the next 5
minutes. Let RV X be the number of hits in t
minutes, what is the pmf in terms of t minutes?
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Distribution ParametersBinomial n
pNegative Binomial r pHypergeometic
n, M, NPoisson ?
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Expected Values Variancespmf E(X)
V(X)Binomial np np(1-p)Negative
Binomial r(1-p) r(1-p) p
p2Hypergeometric nM nM(N-M)(N-n)
N N2 (N-1)Poisson ? ?