# tnorm: The Truncated Normal Distribution In crch: Censored Regression with Conditional Heteroscedasticity

 tnorm R Documentation

## The Truncated Normal Distribution

### Description

Density, distribution function, quantile function, and random generation for the left and/or right truncated normal distribution.

### Usage

```dtnorm(x, mean = 0, sd = 1, left = -Inf, right = Inf, log = FALSE)

ptnorm(q, mean = 0, sd = 1, left = -Inf, right = Inf,
lower.tail = TRUE, log.p = FALSE)

qtnorm(p, mean = 0, sd = 1, left = -Inf, right = Inf,
lower.tail = TRUE, log.p = FALSE)

rtnorm(n, mean = 0, sd = 1, left = -Inf, right = Inf)
```

### Arguments

 `x, q` vector of quantiles. `p` vector of probabilities. `n` number of observations. If `length(n) > 1`, the length is taken to be the number required. `mean` vector of means. `sd` vector of standard deviations. `left` left censoring point. `right` right censoring point. `log, log.p` logical; if TRUE, probabilities p are given as log(p). `lower.tail` logical; if TRUE (default), probabilities are P[X <= x] otherwise, P[X > x].

### Details

If `mean` or `sd` are not specified they assume the default values of `0` and `1`, respectively. `left` and `right` have the defaults `-Inf` and `Inf` respectively.

The truncated normal distribution has density

f(x) = 1/σ φ((x - μ)/σ) / (Φ((right - μ)/σ) - Φ((left - μ)/σ))

for left ≤ x ≤ right, and 0 otherwise.

Φ and φ are the cumulative distribution function and probability density function of the standard normal distribution respectively, μ is the mean of the distribution, and σ the standard deviation.

### Value

`dtnorm` gives the density, `ptnorm` gives the distribution function, `qtnorm` gives the quantile function, and `rtnorm` generates random deviates.

`dnorm`