Lecture 7. Markov Model with Matrixes. Introduction to HMM with the example of DNA analysis. Distributions. Probability density and cumulative distribution functions. Poisson and Normal distributions. Practice: Distributions with Mathematica

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Lecture 7.Markov Model with Matrixes.
Introduction to HMM with the example of DNA
analysis.Distributions. Probability density and
cumulative distribution functions.Poisson and
Normal distributions.Practice Distributions
with Mathematica
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7.1 ReminderWe remind here some facts about the
distributions of discrete and continuous random
variables.
The discrete random distribution can be
characterized by a probability function p(xi)
assigning the probabilities to all possible
values xi of a random variable X. The
probability function should satisfy the following
equations
Compare with the pages 31 of Lect. 1
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Example Suppose that a coin is tossed twice, so
that the sample space is ?HH,HT,TH,TT. Let X
represent a number of heads that can come up.
Find the probability function p(x). As we know,
the probability function is thus given by the
table
x 0 1 2
p(x) 1/4 1/2 1/4
The graphical example presented below represents
a typical way of depicting the probability
distribution for a discrete random variable
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p(xi)
0.3
1
2
3
4
5
6
7
8
9
10
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Possible values of the random variable, xi
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Continuous distribution. Probability density
function (PDF).
Remember For a continuous variable we must
assign to each outcome a probability p(x )0.
Otherwise, we would not be able to fulfill the
requirement 7.1 (the second of three).
A random variable X is said to have a continuous
distribution with density function f(x) if for
all a? b we have
The analogs of Eqs. 7.2 and 7.3 for the
continuous distributions would be
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P(E) is a probability that X belongs to E.
f(x)
P(altXltb)
a
b
Geometrically, P(altXltb) is the area under the
curve f(x) between a and b. The question Can
f(x) exceed 1? Please argue.
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7.2 A few new distributions and their
properties a. Poission distribution Poisson
distribution is one of the most important
discrete distributions. Its probability function
is
Poisson distribution is a limiting case of the
Binomial distribution P(pn,n), with parameters
pn and n such that
In other words, if we have a large number of
independent events with small probability, then
number of occurrences has approximately Poisson
distribution.
Let us introduce now a more intuitive definition
of the Poisson distribution.
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Examples with Poisson distribution 1. Suppose
that the probability of a defect in a foot of
magnetic tape is 0.002. Use the Poisson
distribution to compute the probability that 1500
feet roll will have no defects .
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This example helps to describe the Poisson
Distribution in a new way by noticing that is
the expected (average) value of the defects in
1500 feet of the tape. In other words, the PD
gives the probability of n events happening in
some experiment if the expected (average) number
of the events, , for this particular
experiment, is known. Attention it is important
to understand that is the average (expected)
value for the interval (of time, or space) where
the n questioned events should occur. For
instance, in the former example is the average
value of defects per 1500 feet of tape (but not
per a foot, or per 1000 feet, etc).
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Example 2 An airline company sells 200 tickets
for a plane with 198 seats, knowing that a
probability that a passenger will not show up for
the flight is 0.008. Use the Poisson
approximation to compute the probability that
they will have enough seats for all the
passengers that will show up.
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Solution. p0.012, L0.0122002.4 the average
number (out of 200 passengers ) that wont show
up for the flight. px Exp-2.42.4x/x! Pmore
than 1 person wont show up 1-p0-p1
0.7. In other words, there is (only) a 30 of
chance that they more than 198 passengers will
show up. This is quite a familiar scenario the
company would typically offer you a free
additional ticket for a future flight if you
agree on switching to a later flight. ?
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Example 3 (working in groups) 10 of the tools
produced in a certain manufacturing process turns
out to be defective. Find a probability that in a
sample of ten tools selected at random, exactly 2
will be defective, by using (a) binomial and (b)
Poisson distribution. Open a Mathematica file,
and find the probabilities
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b. The exponential distribution (this is the
continuous distribution, for the continuous
random variable x).
Those who know how to integrate can verify that
(7.8) satisfies (7.5) (the total area under the
curve f(x) equals 1. Note In Matematica, the
integral of a function fx can be found
as Integratefx,x,x1,x2 , ShiftEnter. Here
x1 and x2 are the limits of integration.
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A typical example of the exponential distribution
results from the discussion of the waste products
of the nuclear power plant. If at time t0 there
are N(0) identical unstable particles, and the
number of particles dN(t) decaying in time dt is
proportional to dt and to the number of
particles, then we have
dN(t) - ?N(t)dt This is so called differential
equation. Here is how it is solved with
Mathematica.
DSolvent G nt 0, n0
n0,nt,t nt-gt n0 Exp-Gt
As a result we came up with the exponential
distribution.
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Lets introduce the half-time T , such that
N(T)N0/2. Then we find ?Tln20.693.
c. The standard normal distribution
f(x)(2?)-1/2 exp(-x2/2) (7.12)
A. Using Mathematica, check that this PDF
satisfies the normality condition (7.5). Make a
plot of (7.12). If a random variable y is
related to x as yaxb, how the distribution
function f(y) looks like? (we assume that x is
distributed according to (7.12).
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More generally, X is said to have a normal (?,?2)
distribution if it has density function
f(x)(2? ?2)-1/2 exp-(x- ?)2/2 ?2 (7.12)
?2 is called variance and ? is the mean or
the expectation.
Try to analyze, assigning different numeric
values to ? and ?2 how they affect the shape of
f(x). For instance, how the parameters for the
green and red curves are related? Green and
blue? See the Mathematica file
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7.3 Probability distribution function( also
called cumulative distribution function CDF)
1. Continuous random variable
From the outside, random distributions are well
described by the probability distribution
function (we will use CDF for short) F(x) defined
as
This formula can also be rewritten in the
following very useful form
Question Can F(x) exceed 1? Argue it.
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To see what the distribution functions look like,
we return to our examples.
1. The uniform distribution (7.7)
Using the definition (7.13) and Mathematica, try
to find F(x) for the uniform distribution. Prove
thatF(x)0 for x ?a (x-a)/(b-a) for a ? x ? b
1 for xgtb. Draw the CDF for several a and b.
Consider an important special case a0, b1. How
is it related to the spinner problem? To the
balanced die?
2. The exponential distribution (7.8)
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Use Mathematica to prove that
F(x) 0 for x ? 0 1-exp(-?x) for x gt0.
(7.15)
Lack of memory for the exponential distribution
Suppose that X has an exponential distribution
(7.8). The probability that the event (such as
the radioactive decay) did not happen in t units
of time is P(Xgtt) 1-F(x). According to (7.15)
it results in P(Xgtt) exp(-?t) . Lets find now
a probability that we will have to wait some
additional time s given that we have been waiting
t units of time P(XgttsTgtt) P(X gt ts)/P(X gt
t) exp-?(ts)/ exp-?t) exp-?s. As we
see, the result depends only on s and does not
depend on the previous waiting time. The
probability you must wait additional s units of
time till decay occurs is the same as if you had
not been waiting at all.
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The standard normal distribution
Using Mathematica and Eq. (7.12), find Fx for
the snd. Use NIntegrateft,t,-?,x and
Plot functions.
2. CDF for discrete random variables
For discrete variables the integration is
substituted for summation
It is clear from this formula that if X takes
only a finite number of values, the distribution
function looks like a stairway.
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p(x4)
F(x)
1
p(x3)
p(x2)
p(x1)
x1
x2
x3
x4
x
Draw F(x) for the example in page 7.
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This problem is for your practice Predicting the
rate of mutation based on the Poisson probability
distribution function. The evolutionary process
of amino acid substitutions in proteins is