Probability Distributions

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Normal Distribution

Fig: R functions for normal distribution

pnorm

Probability of obtaining 0 or less in a standard normal distribution:

> pnorm(0) # by default, mean=0 and sd=1

[1] 0.5

Similarly, probability of having 10 or less in a normal distribution with mean 10 and sd 3 -

> pnorm(12, mean=10, sd=3)

[1] 0.7475075

In the following example, lower.tail = F provides the probability of having 1.96 or more:

> pnorm(1.96, lower.tail = F)

[1] 0.0249979

Similarly, probability of obtaining 0 or more in a standard normal distribution:

> pnorm(0, lower.tail=F)

[1] 0.5

68, 95, 99.7 rule -

> pnorm(1)-pnorm(-1)

[1] 0.6826895

> pnorm(2)-pnorm(-2)

[1] 0.9544997

> pnorm(3)-pnorm(-3)

[1] 0.9973002

qnorm

The random variable for which the probability is 0.5 or less in a standard normal distribution -

> qnorm(0.5)

[1] 0

The random variable for which the cumulative probability is 0.025 or less in a standard normal distribution -

> qnorm(0.025)

[1] -1.959964

Using lower.tail=F we can get the upper tail value for the variable for which the probability is 0.025 or less:

> qnorm(0.025, lower.tail=F)

[1] 1.959964

dnorm

dnorm() simply gives the density of a random variable:

> dnorm(0)

[1] 0.3989423

A simple illustration:

> x <- c(-10:10)
> plot(x, dnorm(x, mean=0, sd=3))

rnorm

To generate random numbers from a normal distribution:

> rnorm(10)  # generates 10 standard normal variables

 [1]  0.4543401  0.1647346  2.0118904 -1.0688548 -0.9778855  0.6702705
 [7]  0.3840356 -0.7613346  2.7346792 -0.7315600

Generating variables from a normal distribution with specified mean and sd:

> rnorm(10, mean=150, sd=15)

 [1] 167.5613 136.6430 127.4718 147.3143 127.3961 152.5034 148.5015 134.0661
 [9] 135.6686 143.5774

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Uniform Distribution

Fig: Uniform distribution density function

Fig: Uniform Distribution Cumulative Function

punif

Cumulative distribution function:

> punif(0.5,min=0,max=1)

[1] 0.5

Another example:

> punif(3,min=0, max=20)

[1] 0.15

lower.tail argument can be used to obtain the upper tail’s probability:

> punif(3,min=0, max=20, lower.tail = F)

[1] 0.85

runif

A broadly used function for generating random numbers from a uniform distribution where probability of obtaining any of the numbers is same:

> runif(n=5, min=1, max=10)

[1] 5.540486 1.326721 6.967847 7.076371 1.574161

This generates 5 uniform random numbers between 1 and 10.

Others

R has other functions for uniform distribution which are dunif() and qunif(). Run help() command for details.

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Binomial Distribution

Fig: Binomial Probabiliy Mass Function

pbinom

Cumulative function of a binomial distribution:

> pbinom(5, size = 10, p = 0.4)

[1] 0.8337614

It shows the probability of obtaining 5 or less from a binomial distribution of size 10 and probability of success 0.4.

> x <- 1:10
> plot(x, pbinom(x, 10, 0.4))
> lines(x, pbinom(x, 10, 0.4), col = "blue")

qbinom

It takes the cumulative probability and returns the random variable:

> x <- c(0, 0.3, 0.5, 0.9)
> qbinom(x, 10, 0.4)

[1] 0 3 4 6

rbinom

The following code generates 10 random numbers from binomial distribution with parameters size=10 and probability of success = 0.65:

> rbinom(10, size=10, p=0.65)

 [1] 8 8 6 6 9 7 8 6 7 6

dbinom

This simply shows the density of a value:

> x<- 1:10
> y <- dbinom(x, 10, 0.4)
> plot(x,y)
> lines(x,y)

Logistic distribution

Fig: Logistic distribution

dlogis

dlogis() simply gives the density of a random variable:

> dlogis(x = 0, location = 0, scale = 1)

[1] 0.25

> dlogis(5, 5, 1)

[1] 0.25

> dlogis(0, 0, 3)

[1] 0.08333333

plogis

Cumulative function of a logistic distribution:

> plogis(q = 2, location = 0, scale = 2)

[1] 0.7310586

qlogis

It takes the cumulative probability and returns the random variable:

> prob <- c(0.4, 0.5, 0.75, 0.9)
> qlogis(prob, 0, 3)

[1] -1.216395  0.000000  3.295837  6.591674

> plot(x = qlogis(prob, 0, 3), y = prob,
+      type="h", xlab="Random Variable X",
+      ylab="Probability P(X=k)",
+      font = 2, font.lab = 2)

F distribution

Fig: F distribution

pf

Cumulative distribution:

> pf(2.5, df1=20, df2=20)

[1] 0.9767081

This shows that cumulative probability of F statistic being 2.5 with df 20, 20 is 97.6%

To calculate the upper tail probability:

> pf(2.5, df1=20, df2=20, lower.tail=F)

[1] 0.0232919

qf

> qf(0.05, df1=3, df2=8, lower.tail=F)

[1] 4.066181

This shows, in a F distribution with df 3, 8, the probability will be 0.05 or lower for the values that is greater than 4.06.

Others

R has other functions for F distribution which are df() and rf(). Run help() command for details.