Random Variables and Probability Distributions Modified from a presentation by Carlos J. Rosas-Anderson

Fundamentals of Probability The probability P that an outcome occurs is: The probability P that an outcome occurs is: The sample space is the set of all possible outcomes of an event The sample space is the set of all possible outcomes of an event Example: Visit = {(Capture), (Escape)} Example: Visit = {(Capture), (Escape)}

Axioms of Probability 1. The sum of all the probabilities of outcomes within a single sample space equals one: 2. The probability of a complex event equals the sum of the probabilities of the outcomes making up the event: 3. The probability of 2 independent events equals the product of their individual probabilities:

Probability distributions We use probability distributions because they fit many types of data in the living world We use probability distributions because they fit many types of data in the living world Ex. Height (cm) of Hypericum cumulicola at Archbold Biological Station

Probability distributions Most people are familiar with the Normal Distribution, BUT… Most people are familiar with the Normal Distribution, BUT… …many variables relevant to biological and ecological studies are not normally distributed! …many variables relevant to biological and ecological studies are not normally distributed! For example, many variables are discrete (presence/absence, # of seeds or offspring, # of prey consumed, etc.) For example, many variables are discrete (presence/absence, # of seeds or offspring, # of prey consumed, etc.) Because normal distributions apply only to continuous variables, we need other types of distributions to model discrete variables. Because normal distributions apply only to continuous variables, we need other types of distributions to model discrete variables.

Random variable The mathematical rule (or function) that assigns a given numerical value to each possible outcome of an experiment in the sample space of interest. The mathematical rule (or function) that assigns a given numerical value to each possible outcome of an experiment in the sample space of interest. 2 Types: 2 Types: Discrete random variables Discrete random variables Continuous random variables Continuous random variables

The Binomial Distribution Bernoulli Random Variables Imagine a simple trial with only two possible outcomes: Imagine a simple trial with only two possible outcomes: Success (S) Success (S) Failure (F) Failure (F) Examples Examples Toss of a coin (heads or tails) Toss of a coin (heads or tails) Sex of a newborn (male or female) Sex of a newborn (male or female) Survival of an organism in a region (live or die) Survival of an organism in a region (live or die) Jacob Bernoulli ( )

The Binomial Distribution Overview Suppose that the probability of success is p Suppose that the probability of success is p What is the probability of failure? What is the probability of failure? q = 1 – p q = 1 – p Examples Examples Toss of a coin (S = head): p = 0.5  q = 0.5 Toss of a coin (S = head): p = 0.5  q = 0.5 Roll of a die (S = 1): p =  q = Roll of a die (S = 1): p =  q = Fertility of a chicken egg (S = fertile): p = 0.8  q = 0.2 Fertility of a chicken egg (S = fertile): p = 0.8  q = 0.2

The Binomial Distribution Overview Imagine that a trial is repeated n times Imagine that a trial is repeated n times Examples: Examples: A coin is tossed 5 times A coin is tossed 5 times A die is rolled 25 times A die is rolled 25 times 50 chicken eggs are examined 50 chicken eggs are examined ASSUMPTIONS: ASSUMPTIONS: 1) p is constant from trial to trial 2) the trials are statistically independent of each other

The Binomial Distribution Overview What is the probability of obtaining X successes in n trials? What is the probability of obtaining X successes in n trials? Example Example What is the probability of obtaining 2 heads from a coin that was tossed 5 times? What is the probability of obtaining 2 heads from a coin that was tossed 5 times? P(HHTTT) = (1/2) 5 = 1/32

The Binomial Distribution Overview But there are more possibilities: But there are more possibilities: HHTTTHTHTTHTTHTHTTTH THHTTTHTHTTHTTH TTHHTTTHTH TTTHH P(2 heads) = 10 × 1/32 = 10/32

The Binomial Distribution Overview In general, if n trials result in a series of success and failures, In general, if n trials result in a series of success and failures,FFSFFFFSFSFSSFFFFFSF… Then the probability of X successes in that order is P(X)= q  q  p  q   = p X  q n – X

The Binomial Distribution Overview However, if order is not important, then However, if order is not important, then where is the number of ways to obtain X successes in n trials, and n! = n  (n – 1)  (n – 2)  …  2  1 n!n! X!(n – X)!  p X  q n – X P(X) = n!n! X!(n – X)!

The Binomial Distribution Overview

The Poisson Distribution Overview When there are a large number of trials but a small probability of success, binomial calculations become impractical When there are a large number of trials but a small probability of success, binomial calculations become impractical Example: Number of deaths from horse kicks in the French Army in different years Example: Number of deaths from horse kicks in the French Army in different years The mean number of successes from n trials is λ = np The mean number of successes from n trials is λ = np Example: 64 deaths in 20 years out of thousands of soldiers Example: 64 deaths in 20 years out of thousands of soldiers Simeon D. Poisson ( )

The Poisson Distribution Overview If we substitute λ/n for p, and let n approach infinity, the binomial distribution becomes the Poisson distribution: If we substitute λ/n for p, and let n approach infinity, the binomial distribution becomes the Poisson distribution: P(x) = e -λ λ x x!x!

The Poisson Distribution Overview The Poisson distribution is applied when random events are expected to occur in a fixed area or a fixed interval of time The Poisson distribution is applied when random events are expected to occur in a fixed area or a fixed interval of time Deviation from a Poisson distribution may indicate some degree of non-randomness in the events under study Deviation from a Poisson distribution may indicate some degree of non-randomness in the events under study See Hurlbert (1990) for some caveats and suggestions for analyzing random spatial distributions using Poisson distributions See Hurlbert (1990) for some caveats and suggestions for analyzing random spatial distributions using Poisson distributions

The Poisson Distribution Example: Emission of  -particles Rutherford, Geiger, and Bateman (1910) counted the number of  -particles emitted by a film of polonium in 2608 successive intervals of one-eighth of a minute Rutherford, Geiger, and Bateman (1910) counted the number of  -particles emitted by a film of polonium in 2608 successive intervals of one-eighth of a minute What is n? What is n? What is p? What is p? Do their data follow a Poisson distribution? Do their data follow a Poisson distribution?

The Poisson Distribution Emission of  -particles No.  -particles Observed Over 14 0 Total2608 Calculation of λ: Calculation of λ: λ= No. of particles per interval = 10097/2608 = 3.87 Expected values: Expected values: 2608  P(x) = e (3.87) x x!x! 2608 

The Poisson Distribution Emission of  -particles No.  -particles ObservedExpected Over Total

The Poisson Distribution Emission of  -particles Random eventsRegular eventsClumped events

The Poisson Distribution

Review of Discrete Probability Distributions If X is a discrete random variable, If X is a discrete random variable, What does X ~ Bin(p, n) mean? What does X ~ Bin(p, n) mean? What does X ~ Poisson(λ) mean? What does X ~ Poisson(λ) mean?

The Expected Value of a Discrete Random Variable

The Variance of a Discrete Random Variable

Continuous Random Variables If X is a continuous random variable, then X has an infinitely large sample space If X is a continuous random variable, then X has an infinitely large sample space Consequently, the probability of any particular outcome within a continuous sample space is 0 Consequently, the probability of any particular outcome within a continuous sample space is 0 To calculate the probabilities associated with a continuous random variable, we focus on events that occur within particular subintervals of X, which we will denote as Δx To calculate the probabilities associated with a continuous random variable, we focus on events that occur within particular subintervals of X, which we will denote as Δx

Continuous Random Variables The probability density function (PDF): The probability density function (PDF): To calculate E(X), we let Δx get infinitely small: To calculate E(X), we let Δx get infinitely small:

Uniform Random Variables Defined for a closed interval (for example, [0,10], which contains all numbers between 0 and 10, including the two end points 0 and 10). Defined for a closed interval (for example, [0,10], which contains all numbers between 0 and 10, including the two end points 0 and 10). Subinterval [5,6] Subinterval [3,4] The probability density function (PDF)

Uniform Random Variables For a uniform random variable X, where f(x) is defined on the interval [a,b] and where a<b:

The Normal Distribution Overview Discovered in 1733 by de Moivre as an approximation to the binomial distribution when the number of trials is large Discovered in 1733 by de Moivre as an approximation to the binomial distribution when the number of trials is large Derived in 1809 by Gauss Derived in 1809 by Gauss Importance lies in the Central Limit Theorem, which states that the sum of a large number of independent random variables (binomial, Poisson, etc.) will approximate a normal distribution Importance lies in the Central Limit Theorem, which states that the sum of a large number of independent random variables (binomial, Poisson, etc.) will approximate a normal distribution Example: Human height is determined by a large number of factors, both genetic and environmental, which are additive in their effects. Thus, it follows a normal distribution. Example: Human height is determined by a large number of factors, both genetic and environmental, which are additive in their effects. Thus, it follows a normal distribution. Karl F. Gauss ( ) Abraham de Moivre ( )

The Normal Distribution Overview A continuous random variable is said to be normally distributed with mean  and variance  2 if its probability density function is A continuous random variable is said to be normally distributed with mean  and variance  2 if its probability density function is f(x) is not the same as P(x) f(x) is not the same as P(x) P(x) would be virtually 0 for every x because the normal distribution is continuous P(x) would be virtually 0 for every x because the normal distribution is continuous However, P(x 1 < X ≤ x 2 ) = f(x)dx However, P(x 1 < X ≤ x 2 ) = f(x)dx f (x) = 1  2   (x   ) 2 /2  2 e  x1x1x1x1 x2x2x2x2

The Normal Distribution Overview

Mean changesVariance changes

The Normal Distribution Length of Fish A sample of rock cod in Monterey Bay suggests that the mean length of these fish is  = 30 in. and  2 = 4 in. A sample of rock cod in Monterey Bay suggests that the mean length of these fish is  = 30 in. and  2 = 4 in. Assume that the length of rock cod is a normal random variable  X ~ N(  = 30,  =2) Assume that the length of rock cod is a normal random variable  X ~ N(  = 30,  =2) If we catch one of these fish in Monterey Bay, If we catch one of these fish in Monterey Bay, What is the probability that it will be at least 31 in. long? What is the probability that it will be at least 31 in. long? That it will be no more than 32 in. long? That it will be no more than 32 in. long? That its length will be between 26 and 29 inches? That its length will be between 26 and 29 inches?

The Normal Distribution Length of Fish What is the probability that it will be at least 31 in. long? What is the probability that it will be at least 31 in. long?

The Normal Distribution Length of Fish That it will be no more than 32 in. long? That it will be no more than 32 in. long?

The Normal Distribution Length of Fish That its length will be between 26 and 29 inches? That its length will be between 26 and 29 inches?

Standard Normal Distribution μ=0 and σ 2 =1 μ=0 and σ 2 =1

Useful properties of the normal distribution The normal distribution has useful properties: The normal distribution has useful properties: Can be added: E(X+Y)= E(X)+E(Y) and σ 2 (X+Y)= σ 2 (X)+ σ 2 (Y) Can be added: E(X+Y)= E(X)+E(Y) and σ 2 (X+Y)= σ 2 (X)+ σ 2 (Y) Can be transformed with shift and change of scale operations Can be transformed with shift and change of scale operations

Consider two random variables X and Y Let X~N( μ,σ) and let Y=aX+b where a and b are constants Change of scale is the operation of multiplying X by a constant a because one unit of X becomes “a” units of Y. Shift is the operation of adding a constant b to X because we simply move our random variable X “b” units along the x-axis. If X is a normal random variable, then the new random variable Y created by these operations on X is also a normal random variable.

For X~N( μ,σ) and Y=aX+b E(Y) =aμ+b E(Y) =aμ+b σ 2 (Y)=a 2 σ 2 σ 2 (Y)=a 2 σ 2 A special case of a change of scale and shift operation in which a = 1/σ and b = -1( μ/σ): A special case of a change of scale and shift operation in which a = 1/σ and b = -1( μ/σ): Y = (1/σ)X-(μ/σ) = (X-μ)/σ Y = (1/σ)X-(μ/σ) = (X-μ)/σ This gives E(Y)=0 and σ 2 (Y)=1 This gives E(Y)=0 and σ 2 (Y)=1 Thus, any normal random variable can be transformed to a standard normal random variable. Thus, any normal random variable can be transformed to a standard normal random variable.

The Central Limit Theorem Asserts that standardizing any random variable that itself is a sum or average of a set of independent random variables results in a new random variable that is “nearly the same as” a standard normal one. Asserts that standardizing any random variable that itself is a sum or average of a set of independent random variables results in a new random variable that is “nearly the same as” a standard normal one. So what? The C.L.T allows us to use statistical tools that require our sample observations to be drawn from normal distributions, even though the underlying data themselves may not be normally distributed! So what? The C.L.T allows us to use statistical tools that require our sample observations to be drawn from normal distributions, even though the underlying data themselves may not be normally distributed! The only caveats are that the sample size must be “large enough” and that the observations themselves must be independent and all drawn from a distribution with common expectation and variance. The only caveats are that the sample size must be “large enough” and that the observations themselves must be independent and all drawn from a distribution with common expectation and variance.

Log-normal Distribution X is a log-normal random variable if its natural logarithm, ln(X), is a normal random variable [NOTE: ln(X) is same as log e (X)] X is a log-normal random variable if its natural logarithm, ln(X), is a normal random variable [NOTE: ln(X) is same as log e (X)] Original values of X give a right-skewed distribution (A), but plotting on a logarithmic scale gives a normal distribution (B). Original values of X give a right-skewed distribution (A), but plotting on a logarithmic scale gives a normal distribution (B). Many ecologically important variables are log-normally distributed. Many ecologically important variables are log-normally distributed. SOURCE: Quintana-Ascencio et al. 2006; Hypericum data from Archbold Biological Station A

Log-normal Distribution

Exercise During class we will perform an exercise in R allowing you and us to work with some of these probability distributions! During class we will perform an exercise in R allowing you and us to work with some of these probability distributions!