Density, distribution function, quantile function and random generation for
the truncated Normal distribution with mean equal to `mean`

and
standard deviation equal to `sd`

before truncation, and truncated on
the interval `[lower, upper]`

.

## Usage

```
dtnorm(x, mean = 0, sd = 1, lower = -Inf, upper = Inf, log = FALSE)
ptnorm(
q,
mean = 0,
sd = 1,
lower = -Inf,
upper = Inf,
lower.tail = TRUE,
log.p = FALSE
)
qtnorm(
p,
mean = 0,
sd = 1,
lower = -Inf,
upper = Inf,
lower.tail = TRUE,
log.p = FALSE
)
rtnorm(n, mean = 0, sd = 1, lower = -Inf, upper = Inf)
```

## Arguments

- x, q
vector of quantiles.

- mean
vector of means.

- sd
vector of standard deviations.

- lower
lower truncation point.

- upper
upper truncation point.

- log
logical; if TRUE, return log density or log hazard.

- lower.tail
logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].

- log.p
logical; if TRUE, probabilities p are given as log(p).

- p
vector of probabilities.

- n
number of observations. If

`length(n) > 1`

, the length is taken to be the number required.

## Value

`dtnorm`

gives the density, `ptnorm`

gives the
distribution function, `qtnorm`

gives the quantile function, and
`rtnorm`

generates random deviates.

## Details

The truncated normal distribution has density

$$ f(x, \mu, \sigma) = \phi(x, \mu, \sigma) / (\Phi(u, \mu, \sigma) - \Phi(l, \mu, \sigma)) $$$$ $$ for \(l <= x <= u\), and 0 otherwise.

\(\mu\) is the mean of the original Normal distribution before
truncation,

\(\sigma\) is the corresponding standard deviation,

\(u\) is the upper truncation point,

\(l\) is the lower
truncation point,

\(\phi(x)\) is the density of the
corresponding normal distribution, and

\(\Phi(x)\) is the
distribution function of the corresponding normal distribution.

If `mean`

or `sd`

are not specified they assume the default values
of `0`

and `1`

, respectively.

If `lower`

or `upper`

are not specified they assume the default
values of `-Inf`

and `Inf`

, respectively, corresponding to no
lower or no upper truncation.

Therefore, for example, `dtnorm(x)`

, with no other arguments, is simply
equivalent to `dnorm(x)`

.

Only `rtnorm`

is used in the `msm`

package, to simulate from
hidden Markov models with truncated normal distributions. This uses the
rejection sampling algorithms described by Robert (1995).

These functions are merely provided for completion, and are not optimized
for numerical stability or speed. To fit a hidden Markov model with a
truncated Normal response distribution, use a `hmmTNorm`

constructor. See the `hmm-dists`

help page for further details.

## References

Robert, C. P. Simulation of truncated normal variables. Statistics and Computing (1995) 5, 121--125

## Author

C. H. Jackson chris.jackson@mrc-bsu.cam.ac.uk