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06/05/2008
Jae Hyun Kim
Chapter 1Probability 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
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Content
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
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Discrete Random Variable
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
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Discrete Probability Distributions
A Bernoulli Trial Single trial with two possible outcomes “success” or “failure” Probability of success = p
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One Bernoulli Trial
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
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The Binomial Distribution
Comments 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
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Bernoulli Trail and Binomial Distribution
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
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The Hypergeometric Distribution
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
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The Uniform/Geometric Distribution
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)
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The Poisson Distribution
Mean / Expected Value
Expected Value of g(y)
Example
Linearity Property
In general,
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Mean
Moment r th moment of the probability distribution about
zero
Mean : First moment (r = 1) r th moment about mean
Variance : r = 2
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General Moments
PGF
Used to derive moments Mean
Variance
If two r.v. X and Y have identical probability generating functions, they are identically distributed
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Probability-Generating Function
Probability density function f(x)
Probability
Cumulative Distribution Function
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Continuous Random Variable
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)
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Approximation
Definition
Useful to derive
m’(0) = E[X], m’’(0) = E[X2], m(n)(0) = E[Xn] mgf m(t) = pgf P(et)
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The Moment-Generating Function
Conditional Probability
Bayes’ Formula
Independence
Memoryless Property
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Conditional Probability
Definition
can be considered as function of PY(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
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Entropy
