# Protein-Protein Interaction

Βιοτεχνολογία

2 Οκτ 2013 (πριν από 4 χρόνια και 9 μήνες)

95 εμφανίσεις

06/05/2008

Jae Hyun Kim

Chapter 1

Probability Theory (
i
) : One Random Variable

Bioinformatics Tea Seminar: Statistical Methods in Bioinformatics

Discrete Random Variable

Discrete Probability Distributions

Probability Generating Functions

Continuous Random Variable

Probability Density Functions

Moment Generating Functions

2

Content

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Discrete Random Variable

Numerical quantity that, in some
experiment

(Sample
Space)

that involves some degree of randomness, takes
one value from
some discrete set of possible values
(EVENT)

Sample Space

Set of all outcomes of an experiment (or observation)

For Example,

Flip a coin { H,T }

Toss a die {1,2,3,4,5,6}

Sum of two dice { 2,3,…,12 }

Event

Any subset of outcome

3

Discrete Random Variable

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The probability distribution

Set of values that this random variable can take, together
with their associated probabilities

Example,

Y = total number of heads when flip a coin twice

Probability Distribution Function

Cumulative Distribution Function

4

Discrete Probability Distributions

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A Bernoulli Trial

Single trial with two possible outcomes

“success” or “failure”

Probability of success = p

5

One Bernoulli Trial

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The Binomial Random Variable

The number of success in a fixed number of
n

independent
Bernoulli trials with the same probability of success for
each trial

Requirements

Each trial must result in one of two possible outcomes

The various trials must be independent

The probability of success must be the same on all trials

The number n of trials must be fixed in advance

6

The Binomial Distribution

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Single Bernoulli Trial = special case (n=1) of
Binomial Distribution

Probability p is often an unknown parameter

There is no simple formula for the cumulative
distribution function for the binomial
distribution

There is no unique “binomial distribution,” but
rather a family of distributions indexed by n
and p

7

Bernoulli Trail and Binomial Distribution

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Hypergeometric

Distribution

N objects ( n red, N
-
n white )

m objects are taken at random, without replacement

Y = number of red objects taken

Biological example

N lab mice ( n male, N
-
n female )

m Mutations

The number Y of mutant males:
hypergeometric

distribution

8

The
Hypergeometric

Distribution

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The Uniform Distribution

Same values over the range

The Geometric Distribution

Number of Y Bernoulli trials before but not including the
first failure

Cumulative distribution function

9

The Uniform/Geometric Distribution

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The Poisson Distribution

Event occurs randomly in time/space

For example,

The time between phone calls

Approximation of Binomial Distribution

When

n is large

p is small

np

is moderate

Binomial (n, p, x ) = Poisson (
np
, x) (

=
np
)

10

The Poisson Distribution

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Mean / Expected Value

Expected Value of g(y)

Example

Linearity Property

In general,

11

Mean

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Definition

12

Variance

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Summary

13

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Moment

r
th

moment of the probability distribution about
zero

Mean : First moment (r =
1
)

r
th

Variance : r =
2

14

General Moments

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PGF

Used to derive moments

Mean

Variance

If two
r.v
. X and Y have identical probability
generating functions, they are identically
distributed

15

Probability
-
Generating Function

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Probability density function f(x)

Probability

Cumulative Distribution Function

16

Continuous Random Variable

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Mean

Variance

Mean value of the function g(X)

17

Mean and Variance

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Chebyshev’s

Inequality

Proof

18

Chebyshev’s

Inequality

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Pdf

Mean & Variance

19

The Uniform Distribution

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Pdf

Mean

, Variance

2

20

The Normal Distribution

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Normal Approximation to Binomial

Condition

n is large

Binomial (
n,p,x
) = Normal (

=
np
,

2
=
np
(
1
-
p), x)

Continuity Correction

Normal Approximation to Poisson

Condition

is large

Poisson (

,x) = Normal(

=

,

2
=

, x)

21

Approximation

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Pdf

Cdf

Mean
1
/

, Variance
1
/

2

22

The Exponential Distribution

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Pdf

Mean and Variance

23

The Gamma Distribution

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Definition

Useful to derive

m’(
0
) = E[X], m’’(
0
) = E[X
2
], m
(n)
(
0
) = E[
X
n
]

mgf

m(t) =
pgf

P(e
t
)

24

The Moment
-
Generating Function

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Conditional Probability

Bayes
’ Formula

Independence

Memoryless

Property

25

Conditional Probability

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Definition

can be considered as function of P
Y
(y)

a measure of how close to uniform that distribution
is, and thus, in a sense, of the unpredictability of
any observed value of a random variable having that
distribution.

Entropy
vs

Variance

measure in some sense the uncertainty of the value
of a random variable having that distribution

Entropy : Function of
pdf

Variance : depends on sample values

26

Entropy

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