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Bioinformatics Tea Seminar: Statistical Methods in Bioinformatics. Chapter 1 Probability Theory ( i ) : One Random Variable. 06/05/2008 Jae Hyun Kim. Content. Discrete Random Variable Discrete Probability Distributions Probability Generating Functions Continuous Random Variable I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
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Presentation Transcript Bioinformatics Tea Seminar: Statistical Methods in Bioinformatics

### Chapter 1Probability Theory (i) : One Random Variable

06/05/2008

Jae Hyun Kim Content
• Discrete Random Variable
• Discrete Probability Distributions
• Probability Generating Functions
• Continuous Random Variable
• Probability Density Functions
• Moment Generating Functions

jaekim@ku.edu Discrete Random Variable
• 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

jaekim@ku.edu Discrete Probability Distributions
• 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

jaekim@ku.edu One Bernoulli Trial
• A Bernoulli Trial
• Single trial with two possible outcomes
• “success” or “failure”
• Probability of success = p

jaekim@ku.edu The Binomial Distribution
• 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

jaekim@ku.edu Bernoulli Trail and Binomial Distribution
• 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

jaekim@ku.edu The Hypergeometric 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

jaekim@ku.edu The Uniform/Geometric 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

jaekim@ku.edu The Poisson 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)

jaekim@ku.edu Mean
• Mean / Expected Value
• Expected Value of g(y)
• Example
• Linearity Property
• In general,

jaekim@ku.edu Variance
• Definition

jaekim@ku.edu Summary

jaekim@ku.edu General Moments
• Moment
• r th moment of the probability distribution about zero
• Mean : First moment (r = 1)
• r th moment about mean
• Variance : r = 2

jaekim@ku.edu Probability-Generating Function
• PGF
• Used to derive moments
• Mean
• Variance
• If two r.v. X and Y have identical probability generating functions, they are identically distributed

jaekim@ku.edu Continuous Random Variable
• Probability density function f(x)
• Probability
• Cumulative Distribution Function

jaekim@ku.edu Mean and Variance
• Mean
• Variance
• Mean value of the function g(X)

jaekim@ku.edu Chebyshev’s Inequality
• Chebyshev’s Inequality
• Proof

jaekim@ku.edu The Uniform Distribution
• Pdf
• Mean & Variance

jaekim@ku.edu The Normal Distribution
• Pdf
• Mean , Variance 2

jaekim@ku.edu Approximation
• 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)

jaekim@ku.edu The Exponential Distribution
• Pdf
• Cdf
• Mean 1/, Variance 1/2

jaekim@ku.edu The Gamma Distribution
• Pdf
• Mean and Variance

jaekim@ku.edu The Moment-Generating Function
• Definition
• Useful to derive
• m’(0) = E[X], m’’(0) = E[X2], m(n)(0) = E[Xn]
• mgf m(t) = pgf P(et)

jaekim@ku.edu Conditional Probability
• Conditional Probability
• Bayes’ Formula
• Independence
• Memoryless Property

jaekim@ku.edu Entropy
• 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

jaekim@ku.edu