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Probability Distributions 2014/04/07 Maiko Narahara http://en.wikibooks.org/wiki/R_Programming/ Probability_Distributions

Probability Distributions 2014/04/07 Maiko Narahara

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Page 1: Probability Distributions 2014/04/07 Maiko Narahara

Probability Distributions

2014/04/07Maiko Narahara

http://en.wikibooks.org/wiki/R_Programming/Probability_Distributions

Page 2: Probability Distributions 2014/04/07 Maiko Narahara

Probability density function(PDF)

• A function that defines probabilities of continuous variables• Because a continuous variable is continuous, the probability

of observing the exact value is almost zero.• So, we read the area of a certain range as a probability of

observing any value in the range.• --> Area under the curve must be 1.

Page 3: Probability Distributions 2014/04/07 Maiko Narahara

Probability mass function(PMF)

• A function that defines probabilities for discrete variables.• Unlike continuous variables, the probability of observing exact value is

defined (y-axis = probability). • The sum of y values for all possible x values must be 1.

http://en.wikipedia.org/wiki/File:Binomial_distribution_pmf.svg

Page 4: Probability Distributions 2014/04/07 Maiko Narahara

R functions for probability distributions

• [rdpq]name_of_distribution()– r: random generation• generate random numbers from distribution

– d: density distribution function• returns density for given value

– p: cumulative distribution function• returns cumulative probability• or used when we calculate p value

– q: quantile function• returns values that correspond to given quantiles

Page 5: Probability Distributions 2014/04/07 Maiko Narahara

rnorm

n <- 1000x <- rnorm(n, mean=0, sd=1)hist(x)

Page 6: Probability Distributions 2014/04/07 Maiko Narahara

dnorm

dnorm(x=1, mean=0, sd=1)gives the density that corresponds to the given value x.

Den

sity

X

Page 7: Probability Distributions 2014/04/07 Maiko Narahara

pnormpnorm(q=0, mean=0, sd=1) gives the cumulative probability for the given value of x

How to compute p valueZ-test statistic: 2.5pnorm(2.5, lower.tail=FALSE)

*note: one-tail testCu

mul

ative

pro

babi

lity

X

Page 8: Probability Distributions 2014/04/07 Maiko Narahara

qnormqnorm(0.975) returns x that corresponds to the given quantile value.

This example calculates the upper critical value at alpha=0.05 (two-tail).

Cum

ulati

ve p

roba

bilit

y

X

Page 9: Probability Distributions 2014/04/07 Maiko Narahara

Tips 1Handling vectors

• rnorm(10, mean=1:10, sd=1:10)• rnorm(5, mean=c(1, 1, 2, 2, 2))– # sampling from different distributions

• dnorm(0, mean=1:2)• dnorm(c(0, 1), mean=1:2)– # similarly, qnorm and pnorm can handle vectors

Page 10: Probability Distributions 2014/04/07 Maiko Narahara

Tips 2Drawing curve of d/p function

• Syntax: curve(function, from, to)curve(dnorm, from=-3, to=3) – # draws a nice curve for the standard normal distribution,

• But if you want to change the parameters for the distribution, how to do that?

curve(dnorm, mean=1, sd=2) # does not worka <- function(x) dnorm(x, mean=1, sd=2)curve(a, from=-3, to=5)• Similarly, you can draw a cumulative curvecurve(pnorm, from=-3, to=3)

Page 11: Probability Distributions 2014/04/07 Maiko Narahara

Note about lower.tail=FALSE for discrete distributions

pbinom(1, 5, prob=0.3)--> 0.52822--> include the probability of x=1pbinom(1, 5, prob=0.3, lower.tail=FALSE)--> 0.47178--> does not include x=1Note that setting lower.tail=FALSE equals 1 - pbinom(1, 5, prob=0.3)