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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

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content
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
  • 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
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
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 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
Bernoulli Trail and 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

jaekim@ku.edu

the hypergeometric distribution
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/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
  • 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

slide11
Mean
  • Mean / Expected Value
  • Expected Value of g(y)
    • Example
  • Linearity Property
  • In general,

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variance
Variance
  • Definition

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summary
Summary

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general moments
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
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
Continuous Random Variable
  • Probability density function f(x)
  • Probability
  • Cumulative Distribution Function

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mean and variance
Mean and Variance
  • Mean
  • Variance
  • Mean value of the function g(X)

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chebyshev s inequality
Chebyshev’s Inequality
  • Chebyshev’s Inequality
  • Proof

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the uniform distribution
The Uniform Distribution
  • Pdf
  • Mean & Variance

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the normal distribution
The Normal Distribution
  • Pdf
  • Mean , Variance 2

jaekim@ku.edu

approximation
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
The Exponential Distribution
  • Pdf
  • Cdf
  • Mean 1/, Variance 1/2

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the gamma distribution
The Gamma Distribution
  • Pdf
  • Mean and Variance

jaekim@ku.edu

the moment generating function
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
  • Conditional Probability
  • Bayes’ Formula
  • Independence
  • Memoryless Property

jaekim@ku.edu

entropy
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