a student attempts a multiple choice exam
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Example
A student attempts a multiple choice exam
(options A to F for each question), but
having done no work, selects his answers toeach question by rolling a fair die (A = 1, B =
2, etc.).
If the exam contains 100 questions, what is
the probability of obtaining a mark below 20?
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Simulation
Now, let us simulate a large number of
realisations of students using this random
method of answering multiple choicequestions. We still require the same
Binomial distribution with n=100 and a=
This can be done on R using the command
rbinom.
1
6
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For example, lets simulate 1000 students.
> xsim=rbinom(1000,100,1/6)
> xsim [1] 18 22 9 17 18 20 21 16 8 18 11 16 16 13 16 14 25 15 16 17
[21] 13 25 11 24 17 16 13 21 10 17 18 10 17 18 19 17 19 15 13 12
[41] 15 11 21 23 19 14 19 25 23 19 20 17 17 15 16 14 13 16 17 14
[61] 24 21 19 8 18 20 22 16 15 20 19 17 13 15 13 21 22 12 12 12
[81] 11 14 11 12 16 16 17 21 17 16 17 14 9 17 16 17 12 20 16 17
[101] 18 13 15 16 12 15 17 16 17 26 18 14 21 15 10 23 12 16 16 12
[121] 17 18 22 17 18 14 19 22 13 17 21 15 21 16 17 16 16 28 16 17
[141] 18 19 16 11 14 18 16 18 18 14 20 13 19 19 22 22 13 17 19 17
[161] 18 20 11 22 19 25 15 15 17 18 5 15 14 13 18 15 17 15 20 17
[181] 16 14 23 17 16 10 12 16 21 30 16 13 22 14 15 16 17 14 16 18[201] 14 20 16 19 25 14 15 24 22 19 15 17 22 10 20 13 10 15 14 22
[221] 17 12 16 19 20 17 15 21 14 13 21 11 19 9 21 22 16 13 13 12
[241] 14 13 18 8 14 18 10 16 10 12 21 18 15 17 16 8 19 17 11 18
[261] 23 17 20 16 12 20 11 16 22 17 16 13 22 20 15 15 20 17 22 14
[281] 18 23 18 20 20 16 19 16 15 19 18 17 14 22 15 24 17 15 17 22
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[301] 18 22 10 19 24 21 16 14 11 14 20 15 21 11 17 16 20 19 13 14
[321] 17 17 19 15 17 13 18 23 16 12 25 13 13 21 19 16 20 27 19 18
[341] 18 24 15 23 13 13 14 15 23 13 19 15 11 19 17 12 15 15 17 14
[361] 18 20 17 13 16 14 13 20 18 15 18 16 17 20 14 19 21 12 13 17[381] 22 17 19 16 14 18 16 18 12 16 13 15 16 9 15 16 18 22 14 16
[401] 14 17 12 16 21 16 21 13 14 19 18 18 16 19 17 17 17 13 17 11
[421] 16 16 13 10 26 12 20 17 11 19 18 12 15 14 14 20 15 15 15 11
[441] 18 23 20 23 13 12 18 22 12 16 13 21 22 14 18 21 17 12 19 16
[461] 17 18 15 22 22 20 15 16 13 12 19 22 16 20 19 19 16 8 15 12[481] 29 26 19 16 20 15 11 22 15 20 21 14 16 13 17 15 10 13 17 12
[501] 18 20 17 14 13 19 23 11 27 19 17 16 17 20 21 15 20 20 21 19
[521] 21 16 13 21 16 19 13 9 10 20 12 18 14 13 18 19 22 19 21 18
[541] 6 17 17 19 19 22 23 18 13 12 17 16 21 16 18 21 19 13 22 19
[561] 20 17 18 15 17 15 15 10 18 13 23 17 14 23 22 10 18 11 11 18[581] 16 17 14 13 9 12 14 14 21 23 24 19 12 15 17 18 11 14 19 19
[601] 19 16 17 13 13 15 17 18 17 13 9 19 18 22 17 13 14 22 13 23
[621] 23 19 19 16 24 14 17 18 17 13 16 12 7 15 17 16 18 22 19 15
[641] 16 18 18 13 20 18 12 6 15 11 16 19 12 13 11 17 11 15 11 19
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[661] 17 16 16 21 12 18 20 19 16 14 18 17 16 14 11 17 17 16 17 17
[681] 17 18 16 18 12 18 18 20 19 13 12 16 14 13 13 6 15 12 19 14
[701] 20 17 16 14 21 19 15 26 17 20 12 24 13 11 19 21 18 13 9 16
[721] 9 16 17 16 15 12 11 21 21 13 19 13 13 16 11 17 15 19 22 19[741] 11 13 14 16 20 15 16 12 18 14 12 14 21 12 23 21 19 10 24 17
[761] 17 19 19 15 18 12 14 14 14 20 12 20 12 21 19 20 21 20 17 18
[781] 15 12 16 23 16 16 19 15 12 14 21 25 12 19 20 22 17 16 21 20
[801] 23 24 17 20 17 19 14 22 20 25 10 12 15 16 7 14 14 18 22 10
[821] 15 22 23 18 12 10 14 18 15 15 18 10 21 11 20 15 20 10 13 16[841] 16 17 22 19 19 16 8 20 17 13 21 16 25 16 13 17 14 17 19 21
[861] 17 19 14 22 20 18 14 19 17 23 20 18 14 11 16 18 26 24 24 18
[881] 21 16 23 20 14 16 15 13 14 11 12 13 14 16 18 17 16 17 13 20
[901] 22 8 17 17 16 16 14 22 17 18 18 21 15 11 20 21 18 15 19 21
[921] 16 22 14 12 16 20 16 21 11 13 19 14 23 12 12 17 14 15 26 17[941] 18 14 21 17 14 24 21 12 21 13 20 22 11 20 10 16 16 15 19 13
[961] 16 15 16 17 9 14 11 12 19 17 16 15 21 14 15 14 15 17 15 16
[981] 19 11 15 17 17 17 11 18 21 14 15 17 18 16 11 22 19 16 14 15
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It makes sense now to look at properties of
these 1000 simulations which have been
placed in the vector xsim.
> mean(xsim)
[1] 16.624
> median(xsim)
[1] 17
> sd(xsim)
[1] 3.778479> var(xsim)
[1] 14.2769
>
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Now compare the actual values from the
simulations, with the theoretical values fromthe probability distribution.
SIMULATION THEORETICAL
MEAN 16.624 16.66667
VARIANCE 14.2769 13.88889
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A full summary of the results of the simulation
is given with:
> table(xsim)
xsim
5 6 7 8 9 10 11 12 13 14 15 16 171 3 2 7 10 21 40 57 72 80 82 118 118
18 19 20 21 22 23 24 25 26 27 28 29 30
85 83 61 55 46 25 14 9 6 2 1 1 1>
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A Histogram can also be plotted of this:
> hist(xsim)
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Notice that a BARPLOT of xsim does
NOT produce a useful graph!
> barplot(xsim)
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A barplot of the TABLE of xsim does
work,though.
> barplot(table(xsim))
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Poisson Distribution
The Poisson distribution is used to model thenumber of events occurring within a given time
interval. The formula for the Poisson
probability density (mass) function is
is the shape parameter which indicates the
average number of events in the given time
interval.
( )
!
xe
p x
x
=
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Some events are rather rare - they don't
happen that often. For instance, car
accidents are the exception rather than therule. Still, over a period of time, we can say
something about the nature of rare events.
An example is the improvement of traffic
safety, where the government wants to know
whether seat belts reduce the number ofdeath in car accidents. Here, the Poisson
distribution can be a useful tool to answer
questions about benefits of seat belt use.
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Other phenomena that often follow a Poisson
distribution are death of infants, the number of
misprints in a book, the number of customersarriving, and the number of activations of a
Geiger counter.
The distribution was derived bythe French mathematician
Simon Poisson in 1837, and
the first application was thedescription of the number of
deaths by horse kicking in the
Prussian army.
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Example
Arrivals at a bus-stop follow a
Poisson distribution with an average
of 4.5 every quarter of an hour.
Obtain a barplot of the distribution
(assume a maximum of 20 arrivals in
a quarter of an hour) and calculatethe probability of fewer than 3 arrivals
in a quarter of an hour.
http://images.google.co.uk/imgres?imgurl=http://www.nataliedee.com/painting-busstop.jpg&imgrefurl=http://www.nataliedee.com/newpaintings.php&h=687&w=364&sz=31&tbnid=dac4lfht2FwJ:&tbnh=136&tbnw=72&start=2&prev=/images%3Fq%3Dbusstop%26hl%3Den%26lr%3D -
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The probabilities of 0 up to 2 arrivals
can be calculated directly from theformula
( ) !
xe
p x x
=
4.5 0
4.5(0)0!
ep
=
with =4.5
So p(0) = 0.01111
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Similarly p(1)=0.04999 and p(2)=0.11248
So the probability of fewer than 3 arrivals
is 0.01111+ 0.04999 + 0.11248 =0.17358
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R Code
As with the Binomial distribution, the
codes
dpois and
ppoiswill do the calculations for you.
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> x=dpois(0:20,4.5)
> x[1] 1.110900e-02 4.999048e-02 1.124786e-01 1.687179e-01 1.898076e-01
[6] 1.708269e-01 1.281201e-01 8.236295e-02 4.632916e-02 2.316458e-02
[11] 1.042406e-02 4.264389e-03 1.599146e-03 5.535504e-04 1.779269e-04
[16] 5.337808e-05 1.501258e-05 3.973919e-06 9.934798e-07 2.352979e-07[21] 5.294202e-08
>
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> barplot(x,names=0:20)
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Now check that ppois gives the same answer
(ppois is a cumulative distribution).
> ppois(2,4.5)[1] 0.1735781
>
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Consider a collection of graphs fordifferent values of
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=3
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=4
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=5
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=6
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=10
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In the last case, the probability of 20arrivals is no longer negligible, so
values up to, say, 30 would have to be
considered.
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Properties of Poisson
The mean and variance are both equal to
.
The sum of independent Poisson variables
is a further Poisson variable with mean
equal to the sum of the individual means.
As well as cropping up in the situations
already mentioned, the Poisson distribution
provides an approximation for the Binomial
distribution.
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Approximation:
If n is large and p is small, then the
Binomial distribution with parameters n andp, ( B(n;p) ), is well approximated by the
Poisson distribution with parameter np, i.e.
by the Poisson distribution with the samemean
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Example
Binomial situation, n= 100, p=0.075
Calculate the probability of fewer than10 successes.
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> pbinom(9,100,0.075)[1] 0.7832687
>
This would have been very tricky with
manual calculation as the factorials
are very large and the probabilitiesvery small
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The Poisson approximation to the
Binomial states that will be equal
to np, i.e. 100 x 0.075
so =7.5
> ppois(9,7.5)[1] 0.7764076
>
So it is correct to 2 decimal places.
Manually, this would have been much
simpler to do than the Binomial.
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