dnorm, pnorm, qnorm, rnorm

dnorm()

yields the value of the probability density function (pdf) or the y-value, given a:

• random variable x.
• population mean = \(\mu\)
• standard deviation = \(\sigma\)

Default: mean = 0, standard deviation = 1.

dnorm(-1, mean = 0, sd = 1).

[1] 0.2419707

same result:
dnorm(-1): Here x = -1. Corresponding y value = dnorm(-1).

dnorm(-1):

[1] 0.2419707

pnorm()

\(pnorm(q, \mu = 0, \sigma = 1, lower.tail = TRUE)\)

• Yields the probability under the curve to the left of q.
  • i.e. set lower.tail = TRUE
• \(pnorm(q, \mu = 0, \sigma = 1, lower.tail = TRUE)\) is : \(Pr(Z\le{q})\)
• \(Pr(Z\le{q})\) is the Cumulative Distribution Function (CDF)
• Below \(q = -1\)

q = -1
pnorm(-1): Pr(Z \(\le\) -1)

pnorm(q) 

[1] 0.1586553

lower.tail = TRUE is the default. Same result below.

pnorm(q, lower.tail = TRUE)

[1] 0.1586553

lower.tail = FALSE

Yields the probability under the curve to the RIGHT(i.e. set lower.tail = FALSE) of -1.

q = -1
pnorm(-1, lower.tail = FALSE): Pr(Z \(\ge\) -1)

pnorm(q, lower.tail = FALSE)

[1] 0.8413447

qnorm()

\(qnorm(p, \mu = 0, \sigma = 1, lower.tail = TRUE)\)

• Find the Z-score corresponding to the \(p^{th}\) quantile of the normal distribution

• Yields the z score corresponding to the given probability (area under the curve)

• yields z score (value on z axis corresponding to the given shaded region probability)

• All parameters except the quantile p have a default value

EXAMPLES:

Example 1:

Find \(z\) such that \(P(Z\le {z}) =\) 0.99

Z-score: qnorm(0.99) = 2.3263479

Z-score: z = 2.33

\(P(Z\le\) 2.33) = 0.99

Example 2:

Find z such that \(P(Z\le {z}) =\) 0.8413447

Z-score: qnorm(0.8413447) = 0.9999998

Z-score: z = 1

\(P(Z\le\) 1) = 0.8413447

Example 3:

Find z such that \(P(Z\ge {z}) =\) 0.8413447

Z-score: qnorm(0.8413447, lower.tail = FALSE) = -0.9999998

Z-score: z = -1

\(P(Z\ge\) -1) = 0.8413447

rnorm(n, mean = 0, sd = 1)

Generates n random variable from the normal distribution with mean = 0 and standard deviation = 1.

rnorm(6): yields 6 random variable from the standard normal distribution (mean = 0, s.d. = 1)

rnorm(6)

[1]  0.8217731  1.3921164 -0.4761739  0.6503486  1.3911105 -1.1107889

rnorm(6, mean = 5, sd = 2): yields 6 random variable from the normal distribution with mean = 5, standard deviation = 2

rnorm(6, mean = 5, sd = 2)

[1] 3.278415 2.736523 2.081572 5.159965 6.306409 7.401931

Source: https://www.statology.org/dnorm-pnorm-rnorm-qnorm-in-r/