BScHND IETM Week 910 Some Probability Distributions

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BSc/HND IETM Week 9/10 - Some Probability
Distributions
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When we looked at the histogram a few weeks ago,
we were looking at frequency distributions.
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It is possible to convert such frequency
distributions into probability distributions,
such that the probability of encountering some
particular value (or range of values) of x is
plotted on the vertical axis, rather than the
number of occurrences of that value of x.
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There are a few standard forms of such
distributions, which make analysis rather easy -
so long as the data really do fit the chosen form.
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We shall look at two of these standard forms, the
normal and the negative exponential distributions.
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Probability distributions from frequency
distributions
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Suppose that our previously-mentioned (and,
sadly, hypothetical) optional unit for your
course, Flower Arranging for Engineers, becomes
extremely popular.
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In fact, it becomes so popular that it is studied
by 208 students, from all the various BSc courses
in the School.
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In an effort to analyse the performance of the
students, so as to determine if any improvements
to the unit are required, we might decide to plot
a histogram of the final marks obtained.
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As we know, this is a frequency distribution, and
might be obtained from the following summary of
the students scores, as shown
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Frequency polygonsThe first step in the
conversion is to change from the histogram to
what is called a frequency polygon. This is
simply a line graph, joining the centres of each
of the chosen data intervals.
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At the ends, our frequency polygon reaches the
zero axis as shown, since no student can obtain
less than zero or more than 100 per cent. In
situations when this doesnt apply, it is
conventional to terminate the polygon on the zero
axis, half way through the next interval.
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It is very easy to obtain probability
distributions from diagrams such as those above.
All that is necessary is to divide each frequency
by the total number of (in this case) students,
to obtain the probability of any individual
student, selected at random, obtaining a mark in
a particular range.
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For example, to convert the histogram on page 1,
or the frequency polygon on page 2, into
probability distributions, simply divide every
number on the vertical axis (and therefore also
the numbers written on the plots) by 208.
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Thus, the vertical axes would now be calibrated
in probabilities from zero to 53/208 0.255.
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The probability of any given student obtaining a
mark in the range 40 to 49.9 per cent will be
47/208 0.226. The probability of a student
scoring 90 per cent or more will be
3/208  0.0144, etc.
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The normal distribution

• It is not very surprising that the marks
  distribution (frequency or probability) looks
  like the diagrams above.

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In a fair examination, taken by a large number of
students, we would expect that only a few
students would obtain either abysmally low marks
or astronomically high marks.
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We would expect the majority of marks to be
somewhere in the middle, with a tail at both
the low and the high ends of the range.
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We would expect the majority of marks to be
somewhere in the middle, with a tail at both
the low and the high ends of the range. This is
what we see above.
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Several real-life situations fit this general
form of distribution, where it is most likely
that results will be clustered around the centre
of some range, with outlying values tailing off
towards the ends of the range.
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Wisniewski, in his Foundation text, uses an
example based on the distributions of the weights
of breakfast cereal packed by machines into boxes.
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There should always ideally be the stated amount
in a box but, inevitably, some boxes will be
lighter, and some heavier. There will be the odd
rogue boxes a long way from the mean.
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To make it easier to cope with such situations,
they are often assumed to fit a standardised
probability distribution, called the normal
distribution.
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By doing this, it is possible to use standard
printed tables to make predictions such as (for
example), how many students would be expected to
score less than 40 per cent
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To allow standard tables to be used, we need to
assume a certain fixed shape of probability
distribution, and we also need to define it in
terms of mean and standard deviation.
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We cannot define it in terms of actual data
values (e.g. examination marks, or weight of
cereal in a box), otherwise we would need a
different set of tables for every new problem.
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The normal distribution curve is actually defined
by a rather unpleasant formula (but we dont need
to use it, as we are going to use tables which
have been derived from it by someone else).
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If the variable in which we are interested is x
(e.g. a mark in per cent, or the weight of
cereal in a box in kg), the mean value of x is
and the standard deviation of the data set is
?x,
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then the normal distribution curve is defined by
the probability that x will take a particular
value (P(x)) obeying the following relationship
(I believe there is an error in Wisniewskis
version)
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The resulting plot of P(x) as x varies is a
bell-shaped curve, as shown in the next slide.
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Notes1. The x axis is in STANDARD
DEVIATIONS2. The total area under the graph is 1
unit.3. The area under the graph between two
values of x gives the probability that the
quantity will be between those values.
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ExampleSay that a large set of examination
results has a mean of 55 per cent, and a standard
deviation of 15 per cent.
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How many students would we expect to fail the
examination (if we define a failure as obtaining
less than 40 per cent), and how many students
would we expect to get a first-class result
(defined as obtaining 70 per cent or more)?
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X 1.0 (1 SD from mean)First Probability
0.1587Fail Also 0.1587 !
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The negative exponential distributionTo cover a
wider range of real-world situations, more
standardised probability distributions are
required.
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The other one we shall briefly look at is the
negative-exponential distribution. This is also
sometimes called a failure-rate curve, because
it tends to describe how components fail with
time.
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If a certain number of components is manufactured
and put into service, it is reasonable to assume
that they will all eventually fail.
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If a certain number of components is manufactured
and put into service, it is reasonable to assume
that they will all eventually fail. The
probability of any one of the components failing
during a given time period might well depend on
how many components are left in service.
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Choose to measure time t in the best units for
the problem (seconds, months, years, etc.).
Technically, the unit chosen should be short
compared with the expected lifetime of a
component, so that any given component is
expected to last for many time units.
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Let ? be the failure rate, that is, the
proportion of components expected to fail in one
time unit. This means that ? must have
dimensions of (1/time). In the example above,
we said that 1 per cent of components might fail
in three years so, in that case, the failure rate
? ? 0.01/3 (proportion per year).
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This can also be viewed as a probability - there
is a probability of 0.01/3 that any given
component will fail in a given period of one
year.
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Therefore, to find the proportion of components
expected to fail over a time t (measured in our
chosen units), we need the quantity ?t. This is
now dimensionless - it is actually the
probability that any given component will fail
over the stated time period.
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We can now state the rate of change of the number
of components as follows (it is negative, because
the number decreases as time passes)
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This is called a differential equation and would
normally be written
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in which the quantity dN / dt is to be
interpreted as the rate of change of N as the
time progresses.
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It turns out that
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We can plot this negative exponential function as
the following curve relating the remaining number
of components n to time
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The End