Distributions in R

Installation of R

Download R from https://cloud.r-project.org/ Choose different links based on your system
- For Windows, click base to download and install
- For Mac, click the pkg file to download and install

Installation of Rstudio

Rstudio is an excellent interface for R.

Download Rstudio https://www.rstudio.com/products/rstudio/download/

Start Rstudio

You can directly run functions in Rstudio. For example, type \(1+1\) and click enter
A R script file can be created, which can be saved locally and you can use it later.

Discrete distributions

binomial distribution

\(Binomial(n,p)\)

dbinom: \(P(Y=x)\)
pbinom: \(P(Y\leq y)\)
Examples in the notes

dbinom(3,size = 10,prob = 0.3)

## [1] 0.2668279

pbinom(3,size = 10,prob = 0.3)

## [1] 0.6496107

1 - pbinom(2,10,0.3)

## [1] 0.6172172

pbinom(2,10,0.3)

## [1] 0.3827828

1 - pbinom(3,10,0.3)

## [1] 0.3503893

Geometric distribution

\(Geom(p)\). In R, the random variables is defined as the number of failures before the first success

dgeom: \(P(Y=x)\)
pgeom: \(P(Y\leq x)\)
Examples in the notes

1 - pgeom(2 -1, 0.35)

## [1] 0.4225

dgeom(4-1, 0.35)

## [1] 0.09611875

pgeom(5-1, 0.35)

## [1] 0.8839709

Negative Binomial

\(NB(r,p)\). In R, the random variables is defined as the number of failures before the rth success

dnbinom: \(P(Y=y)\)
pnbinom: \(P(Y\leq y)\)
Examples in the notes

dnbinom(5-3,size = 3,prob = 0.7)

## [1] 0.18522

pnbinom(q = 5-3,size = 3,prob = 0.7)

## [1] 0.83692

Hypergeometric

\(Hyper(N,r,n)\).

dhyper: \(P(Y=y)\)
phyper: \(P(Y\leq y)\)
Examples in the notes.

dhyper(6, 6, 36, 6)

## [1] 1.906292e-07

dhyper(4, 6, 36, 6)

## [1] 0.001801446

1 - phyper(2, 6, 36, 6)

## [1] 0.02906466

Poisson

\(Poisson(\lambda)\)

dpois: \(P(X=x)\)
ppois: \(P(X\leq x)\)
\(P(X=4)\) and \(P(X\leq 4)\) with \(\lambda = 2\)

?dpois
dpois(x = 4,lambda = 2)

## [1] 0.09022352

ppois(q = 4,lambda = 2)

## [1] 0.947347

Continuous distributions

Uniform

\(Uniform(\theta_1, \theta_2)\)

For example, \(\theta_1 = 1\) and \(\theta_2 = 2\)

dunif: probability density function
punif: \(P(X\leq x)\)
qunif: \(P(X\leq q) = p\) pth quantile

dunif(x = 1.5,min = 1,max = 2)

## [1] 1

punif(q = 1.5,min = 1,max = 2)

## [1] 0.5

qunif(p = 0.5,min = 1,max = 2)

## [1] 1.5

Normal

\(Normal(\mu, \sigma^2)\)

For example, \(\mu = 1\) and \(\sigma = 2\)

dnorm: probability density function
pnorm: \(P(X\leq x)\)
qnorm: \(P(X\leq q) = p\) pth quantile

dnorm(x = 1,mean = 1,sd = 2)

## [1] 0.1994711

pnorm(q = 1.5,mean = 1,sd = 2)

## [1] 0.5987063

qnorm(p = 0.5,mean = 1,sd = 2)

## [1] 1

N(0,1), N(1,1), N(-1,1)

N(0,1), N(0,\(2^2\)), N(0,\(0.5^2\))

Example 4.8: Standard normal distribution \(Z\)

    # P(Z > 2)
    1 - pnorm(2)

## [1] 0.02275013

    # P(Z < 2)
    pnorm(2)

## [1] 0.9772499

    # P(Z < -2)
    pnorm(-2)

## [1] 0.02275013

    # P(Z > -2)
    1 - pnorm(-2)

## [1] 0.9772499

    # P(-2 < Z <2)
    pnorm(2) - pnorm(-2)

## [1] 0.9544997

    2*(1 - pnorm(-2))

## [1] 1.9545

    # P(0 < Z < 1.73)
    pnorm(1.73) - pnorm(0)

## [1] 0.4581849

Example of general \(\mu\) and \(\sigma\)

    # Y ~ N(2,4); P(0 < Y <4) = P(-1 <Z < 1)
    pnorm(1) - pnorm(-1)

## [1] 0.6826895

    pnorm(4,mean = 2,sd = 2) - pnorm(0,mean = 2,sd = 2)

## [1] 0.6826895

    # Y ~ N(5,1); P(Y > 7.32) = P(Z > 2.32)
    1- pnorm(2.32)

## [1] 0.01017044

    1 - pnorm(7.32,mean = 5,sd = 1)

## [1] 0.01017044

    # Y ~ N(0,25); P(Y < 7) = P(Z < 7/5)
    pnorm(7/5)

## [1] 0.9192433

    pnorm(7,mean = 0,sd = 5)

## [1] 0.9192433

quantiles

    # Standard normal distribution
    qnorm(0.95)

## [1] 1.644854

    qnorm(0.1)

## [1] -1.281552

    # Y ~ N(75,100)
    75 + 10 * qnorm(0.97)

## [1] 93.80794

    qnorm(0.97,mean = 75,sd = 10)

## [1] 93.80794

Empirical Rule

library(ggplot2)
ggplot(data.frame(x = c(-4, 4)), aes(x)) +
 stat_function(fun = dnorm, geom = "line") +
  geom_vline(xintercept = c(-1,1),color= "red")

pnorm(1) - pnorm(-1)

## [1] 0.6826895

ggplot(data.frame(x = c(-4, 4)), aes(x)) +
 stat_function(fun = dnorm, geom = "line") +
  geom_vline(xintercept = c(-2,2),color= "blue")

pnorm(2) - pnorm(-2)

## [1] 0.9544997

ggplot(data.frame(x = c(-4, 4)), aes(x)) +
 stat_function(fun = dnorm, geom = "line") +
    geom_vline(xintercept = c(-3,3),color= "purple")

pnorm(3) - pnorm(-3)

## [1] 0.9973002

Gamma

dgamma: probability density function
pgamma: \(P(X\leq x)\)
qgamma: \(P(X\leq q) = p\) pth quantile

\(Gamma(\alpha = 1.5, \beta = 2.5)\)

dgamma(x = 1.5,shape = 1.5,scale = 2.5)

## [1] 0.1918731

pgamma(q = 1.5,shape= 1,scale = 2.5)

## [1] 0.4511884

qgamma(p = 0.5,shape = 1.5, scale = 2.5)

## [1] 2.957467

\(\alpha = 1,2,4\), \(\beta = 1\)

\(\alpha = 1,2,4\), \(\beta = 2\)

\(\alpha = 1\), \(\beta = 1,2,4\)

\(\alpha = 2\), \(\beta = 1,2,4\)

\(\alpha = 4\), \(\beta = 1,2,4\)

Beta

dbeta: probability density function
pbeta: \(P(X\leq x)\)
qbeta: \(P(X\leq q) = p\) pth quantile

\(Beta(\alpha = 1.5, \beta = 2.5)\)

dbeta(x = 0.6,shape1 = 1.5,shape2  = 2.5)

## [1] 0.9980119

pbeta(q = 0.6,shape1 = 1.5,shape2  = 2.5)

## [1] 0.8260723

qbeta(p = 0.5,shape1 = 1.5,shape2  = 2.5)

## [1] 0.3524523

\(\alpha = 1,2,4\), \(\beta = 1,2,4\)

\(\alpha = 2\), \(\beta = 2,4,6\)

\(\alpha = 2,4,6\), \(\beta = 2\)

chi-square

dchisq: probability density function
pchisq: \(P(X\leq x)\)
qchisq: \(P(X\leq q) = p\) pth quantile

\(\chi^2(v = 2)\)

dchisq(x = 2.5,df = 2)

## [1] 0.1432524

pchisq(q = 2.5,df = 2)

## [1] 0.7134952

qchisq(p = 0.5,df = 2)

## [1] 1.386294

\(v = 1,2,4\)

exponential

dexp: probability density function
pexp: \(P(X\leq x)\)
qexp: \(P(X\leq q) = p\) pth quantile

\(Exp(\beta = 2)\)

dexp(x = 2.5,1/2)

## [1] 0.1432524

pexp(q = 2.5,1/2)

## [1] 0.7134952

qexp(p = 0.5,1/2)

## [1] 1.386294

\(\beta = 1,2,4\)