Formal Modeling in Cognitive Science 1 Special Probability - PowerPoint PPT Presentation

Special Probability Distributions Special Probability Distributions Special Probability Densities Special Probability Densities Formal Modeling in Cognitive Science 1 Special Probability Distributions Uniform Distribution Lecture 23: Special Distributions and Densities Binominal Distribution Steve Renals (notes by Frank Keller) 2 Special Probability Densities Uniform Distribution School of Informatics University of Edinburgh Exponential Distribution s.renals@ed.ac.uk Normal Distribution 5 March 2007 Steve Renals (notes by Frank Keller) Formal Modeling in Cognitive Science 1 Steve Renals (notes by Frank Keller) Formal Modeling in Cognitive Science 2 Special Probability Distributions Uniform Distribution Special Probability Distributions Uniform Distribution Special Probability Densities Binominal Distribution Special Probability Densities Binominal Distribution Uniform Distribution Binominal Distribution Definition: Uniform Distribution Often we are interested in experiments with repeated trials: A random variable X has a discrete uniform distribution iff its assume there is a fixed number of trials; probability distribution is given by: each of the trial can have two outcomes (e.g., success and f ( x ) = 1 failure, head and tail); k for x = x 1 , x 2 , . . . , x k the probability of success and failure is the same for each trial: where x i � = x j when i � = j . θ and 1 − θ ; the trials are all independent. The mean and variance of the uniform distribution are: Then the probability of getting x successes in n trials in a given k k � n order is θ x (1 − θ ) n − x . And there are � different orders, so the x i · 1 ( x i − µ ) 2 1 σ 2 = � � x µ = � n � θ x (1 − θ ) n − x . overall probability is k k x i =1 i =1 Steve Renals (notes by Frank Keller) Formal Modeling in Cognitive Science 3 Steve Renals (notes by Frank Keller) Formal Modeling in Cognitive Science 4

Special Probability Distributions Uniform Distribution Special Probability Distributions Uniform Distribution Special Probability Densities Binominal Distribution Special Probability Densities Binominal Distribution Binominal Distribution Binominal Distribution 1 Definition: Binomial Distribution 0.9 A random variable X has a binominal distribution iff its probability 0.8 distribution is given by: 0.7 � n � θ x (1 − θ ) n − x for x = 0 , 1 , 2 , . . . , n b(x; 12, 0.5) b ( x ; n , θ ) = 0.6 x 0.5 0.4 Example 0.3 The probability of getting five heads and seven tail in 12 flips of a balanced coin is: 0.2 b (5; 12 , 1 � 12 � (1 2) 5 (1 − 1 0.1 2) 12 − 5 2) = 5 0 1 1 2 3 4 5 6 7 8 9 10 11 12 x Steve Renals (notes by Frank Keller) Formal Modeling in Cognitive Science 5 Steve Renals (notes by Frank Keller) Formal Modeling in Cognitive Science 6 Uniform Distribution Special Probability Distributions Uniform Distribution Special Probability Distributions Exponential Distribution Special Probability Densities Binominal Distribution Special Probability Densities Normal Distribution Binominal Distribution Uniform Distribution If we invert successes and failures (or heads and tails), then the Definition: Uniform Distribution probability stays the same. Therefore: A random variable X has a continuous uniform distribution iff its Theorem: Binomial Distribution probability density is given by: b ( x ; n , θ ) = b ( n − x ; n , 1 − θ ) 1 � for α < x < β β − α u ( x ; α, β ) = 0 elsewhere Two other important properties of the binomial distribution are: Theorem: Binomial Distribution The mean and variance of the uniform distribution are: The mean and the variance of the binomial distribution are: µ = α + β σ 2 = 1 12( α − β ) 2 σ 2 = n θ (1 − θ ) 2 µ = n θ and Steve Renals (notes by Frank Keller) Formal Modeling in Cognitive Science 7 Steve Renals (notes by Frank Keller) Formal Modeling in Cognitive Science 8

Uniform Distribution Uniform Distribution Special Probability Distributions Special Probability Distributions Exponential Distribution Exponential Distribution Special Probability Densities Special Probability Densities Normal Distribution Normal Distribution Uniform Distribution Exponential Distribution Definition: Exponential Distribution 0.6 A random variable X has an exponential distribution iff its 0.5 probability density is given by: 0.4 � 1 θ e − x /θ for x > 0 g ( x ; θ ) = 0.3 0 elsewhere 0.2 The mean and variance of the exponential distribution are: 0.1 σ 2 = θ 2 µ = θ 0 1 1.5 2 2.5 3 3.5 4 x Uniform distribution for α = 1 and β = 4. Steve Renals (notes by Frank Keller) Formal Modeling in Cognitive Science 9 Steve Renals (notes by Frank Keller) Formal Modeling in Cognitive Science 10 Uniform Distribution Uniform Distribution Special Probability Distributions Special Probability Distributions Exponential Distribution Exponential Distribution Special Probability Densities Special Probability Densities Normal Distribution Normal Distribution Exponential Distribution Normal Distribution Definition: Normal Distribution A random variable X has a normal distribution iff its probability 0.35 density is given by: 0.3 1 σ ) 2 for − ∞ < x < ∞ e − 1 2 ( x − µ √ 0.25 n ( x ; µ, σ ) = σ 2 π 0.2 Normally distributed random variables are ubiquitous in 0.15 probability theory; 0.1 many measurements of physical, biological, or cognitive 0.05 processes yield normally distributed data; such data can be modeled using a normal distributions 0 1 2 3 4 5 (sometimes using mixtures of several normal distributions); x Exponential distribution for θ = { 0 . 5 , 1 , 2 , 3 } . also called the Gaussian distribution. Steve Renals (notes by Frank Keller) Formal Modeling in Cognitive Science 11 Steve Renals (notes by Frank Keller) Formal Modeling in Cognitive Science 12

Uniform Distribution Uniform Distribution Special Probability Distributions Special Probability Distributions Exponential Distribution Exponential Distribution Special Probability Densities Special Probability Densities Normal Distribution Normal Distribution Standard Normal Distribution Normal Distribution Definition: Standard Normal Distribution 0.4 The normal distribution with µ = 0 and σ = 1 is referred to as the standard normal distribution: 1 0.3 e − 1 2 x 2 √ n ( x ; 0 , 1) = 2 π 0.2 Theorem: Standard Normal Distribution If a random variable X has a normal distribution, then: 0.1 P ( | x − µ | < σ ) = 0 . 6826 P ( | x − µ | < 2 σ ) = 0 . 9544 0 -4 -2 0 2 4 This follows from Chebyshev’s Theorem (see previous lecture). x Steve Renals (notes by Frank Keller) Formal Modeling in Cognitive Science 13 Steve Renals (notes by Frank Keller) Formal Modeling in Cognitive Science 14 Uniform Distribution Uniform Distribution Special Probability Distributions Special Probability Distributions Exponential Distribution Exponential Distribution Special Probability Densities Special Probability Densities Normal Distribution Normal Distribution Normal Distribution Summary The uniform distribution assigns each value the same Theorem: Z -Scores probability; If a random variable X has a normal distribution with the mean µ and the standard deviation σ then: The binomial distributions models an experiment with a fixed number of independent binary trials, each with the same Z = X − µ probability; σ The normal distribution models the data generated by has the standard normal distribution. measurements of physical, biological, or cognitive processes; Z -scores can be used to convert a normal distribution into the This conversion is often used to make results obtained by different standard normal distribution. experiments comparable: convert the distributions to Z -scores. Steve Renals (notes by Frank Keller) Formal Modeling in Cognitive Science 15 Steve Renals (notes by Frank Keller) Formal Modeling in Cognitive Science 16