Triangular {EnvStats} | R Documentation |
The Triangular Distribution
Description
Density, distribution function, quantile function, and random generation
for the triangular distribution with parameters min
, max
,
and mode
.
Usage
dtri(x, min = 0, max = 1, mode = 1/2)
ptri(q, min = 0, max = 1, mode = 1/2)
qtri(p, min = 0, max = 1, mode = 1/2)
rtri(n, min = 0, max = 1, mode = 1/2)
Arguments
x |
vector of quantiles. Missing values ( |
q |
vector of quantiles. Missing values ( |
p |
vector of probabilities between 0 and 1. Missing values ( |
n |
sample size. If |
min |
vector of minimum values of the distribution of the random variable.
The default value is |
max |
vector of maximum values of the random variable.
The default value is |
mode |
vector of modes of the random variable.
The default value is |
Details
Let be a triangular random variable with parameters
min=
,
max=
, and
mode=
.
Probability Density and Cumulative Distribution Function
The density function of is given by:
| | for |
| for |
|
where .
The cumulative distribution function of is given by:
| | for |
| for |
|
where .
Quantiles
The quantile of
is given by:
| | for |
| for |
|
where .
Random Numbers
Random numbers are generated using the inverse transformation method:
where is a random deviate from a uniform
distribution.
Mean and Variance
The mean and variance of are given by:
Value
dtri
gives the density, ptri
gives the distribution function,
qtri
gives the quantile function, and rtri
generates random
deviates.
Note
The triangular distribution is so named because of the shape of its probability
density function. The average of two independent identically distributed
uniform random variables with parameters min=
and
max=
has a triangular distribution with parameters
min=
,
max=
, and
mode=
.
The triangular distribution is sometimes used as an input distribution in probability risk assessment.
Author(s)
Steven P. Millard (EnvStats@ProbStatInfo.com)
References
Forbes, C., M. Evans, N. Hastings, and B. Peacock. (2011). Statistical Distributions. Fourth Edition. John Wiley and Sons, Hoboken, NJ.
Johnson, N. L., S. Kotz, and N. Balakrishnan. (1995). Continuous Univariate Distributions, Volume 2. Second Edition. John Wiley and Sons, New York.
See Also
Uniform, Probability Distributions and Random Numbers.
Examples
# Density of a triangular distribution with parameters
# min=10, max=15, and mode=12, evaluated at 12, 13 and 14:
dtri(12:14, 10, 15, 12)
#[1] 0.4000000 0.2666667 0.1333333
#----------
# The cdf of a triangular distribution with parameters
# min=2, max=7, and mode=5, evaluated at 3, 4, and 5:
ptri(3:5, 2, 7, 5)
#[1] 0.06666667 0.26666667 0.60000000
#----------
# The 25'th percentile of a triangular distribution with parameters
# min=1, max=4, and mode=3:
qtri(0.25, 1, 4, 3)
#[1] 2.224745
#----------
# A random sample of 4 numbers from a triangular distribution with
# parameters min=3 , max=20, and mode=12.
# (Note: the call to set.seed simply allows you to reproduce this example.)
set.seed(10)
rtri(4, 3, 20, 12)
#[1] 11.811593 9.850955 11.081885 13.539496