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Std normal distribution density A

R computations with normal distributions

There are various R functions which are useful for computation with normal distributions, such as pnorm( ), qnorm( ), and dnorm( ).

The pnorm( ) function gives the cumulative distribution function, and the alphabet ‘p’ stands for probability.  The qnorm( ) is for quantiles whilst the dnorm( ) function, the density.

Let’s use the statistical notation for normal distribution: X ~ N(µ,sigma2).  We shall illustrate the usage of these R functions.

R function pnorm( )

For example, let X ~ N(8,4), then

(a)  the probability P(X < 2) can be computed via pnorm( ) in several different ways:

> pnorm(2,mean=8,sd=2)  #P(X<=2) in N(8,4)

[1] 0.001349898

> pnorm(2,8,2)  #P(X<=2) in N(8,4) simplified

[1] 0.001349898

(b)  the probability P(X < 1.96) for x ~ N(8,4) by R language is:

> pnorm(1.96,8,2)  #P(X<=1.96) in N(8,4)

[1] 0.001263873

Remember that for f(1.96) = 0.975 and f(1.645) = 0.950, respectively from the statistics table, the R gives us the same answers:

> pnorm(1.96,0,1)  #P(X<=1.96) in N(0,1)

[1] 0.9750021

> pnorm(1.645,0,1)  #P(X<=1.645) in N(0,1)

[1] 0.9500151

> pnorm(1.645)  #P(X<=1.645) in N(0,1) simplified

[1] 0.9500151


And, when P(X < -1.645), the R result indicates the area on the left hand side of the normal distribution curve:

> pnorm(-1.645)  #P(X<=-1.645) in N(0,1) simplified

[1] 0.04998491


 R function qnorm( )

In layman’s language, a quantile is where a series of sample data is sub-divided into equal proportions. In statistics, we divide a probability distribution into areas of equal probability. The simplest division that can be envisioned is into two equal halves, i.e., 50%.

The R function: qnorm( ) is used to compute the quantiles for the standard normal distribution using its density function f.

For example,

> qnorm(0.95)  #95.0% quantile of N(0,1)

[1] 1.644854

> qnorm(0.975)  #97.5% quantile of N(0,1)

[1] 1.959964


 R function dnorm( )

The density of a Gaussian formulae for normal distribution can be shown to be close to 0.4 when x = 0.

The R function dnorm(0) indeed gives the same result as below:

> dnorm(0)  # Density of N(0,1) evaluated at x= 0

[1] 0.3989423


Further remarks

Like pnorm( ), the functions qnorm( ) and dnorm( ) can also be used for normal distributions with non-zero mean and non-zero standard deviation or variance, simply by supplying the mean and standard deviation as extra arguments.

For example, for the N(8,4) distribution,  the results are self-explanatory:

> qnorm(0.975,8,2)  # 97.5% quantile of N(8,4)

[1] 11.91993

> dnorm(1,8,2)  #Density of N(8,4) at x=1

[1] 0.0004363413

> dnorm(4,8,2)  #Density of N(8,4) at x=4

[1] 0.02699548



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