## 19.1 Normal distribution

### 19.1.1 Probability density function

If $$\mu \in \mathbb{R}$$ and $$\sigma \in \mathbb{R}^+$$, then for $$y \in \mathbb{R}$$, $\text{Normal}(y|\mu,\sigma) = \frac{1}{\sqrt{2 \pi} \ \sigma} \exp\left( - \, \frac{1}{2} \left( \frac{y - \mu}{\sigma} \right)^2 \right) \!.$

### 19.1.2 Sampling statement

y ~ normal(mu, sigma)

Increment target log probability density with normal_lupdf(y | mu, sigma).
Available since 2.0

### 19.1.3 Stan functions

real normal_lpdf(reals y | reals mu, reals sigma)
The log of the normal density of y given location mu and scale sigma
Available since 2.12

real normal_lupdf(reals y | reals mu, reals sigma)
The log of the normal density of y given location mu and scale sigma dropping constant additive terms.
Available since 2.25

real normal_cdf(reals y, reals mu, reals sigma)
The cumulative normal distribution of y given location mu and scale sigma; normal_cdf will underflow to 0 for $$\frac{{y}-{\mu}}{{\sigma}}$$ below -37.5 and overflow to 1 for $$\frac{{y}-{\mu}}{{\sigma}}$$ above 8.25; the function Phi_approx is more robust in the tails, but must be scaled and translated for anything other than a standard normal.
Available since 2.0

real normal_lcdf(reals y | reals mu, reals sigma)
The log of the cumulative normal distribution of y given location mu and scale sigma; normal_lcdf will underflow to $$-\infty$$ for $$\frac{{y}-{\mu}}{{\sigma}}$$ below -37.5 and overflow to 0 for $$\frac{{y}-{\mu}}{{\sigma}}$$ above 8.25; log(Phi_approx(...)) is more robust in the tails, but must be scaled and translated for anything other than a standard normal.
Available since 2.12

real normal_lccdf(reals y | reals mu, reals sigma)
The log of the complementary cumulative normal distribution of y given location mu and scale sigma; normal_lccdf will overflow to 0 for $$\frac{{y}-{\mu}}{{\sigma}}$$ below -37.5 and underflow to $$-\infty$$ for $$\frac{{y}-{\mu}}{{\sigma}}$$ above 8.25; log1m(Phi_approx(...)) is more robust in the tails, but must be scaled and translated for anything other than a standard normal.
Available since 2.15

R normal_rng(reals mu, reals sigma)
Generate a normal variate with location mu and scale sigma; may only be used in transformed data and generated quantities blocks. For a description of argument and return types, see section vectorized PRNG functions.
Available since 2.18

### 19.1.4 Standard normal distribution

The standard normal distribution is so-called because its parameters are the units for their respective operations—the location (mean) is zero and the scale (standard deviation) one. The standard normal is parameter-free, and the unit parameters allow considerable simplification of the expression for the density. $\text{StdNormal}(y) \ = \ \text{Normal}(y \mid 0, 1) \ = \ \frac{1}{\sqrt{2 \pi}} \, \exp \left( \frac{-y^2}{2} \right)\!.$ Up to a proportion on the log scale, where Stan computes, $\log \text{Normal}(y \mid 0, 1) \ = \ \frac{-y^2}{2} + \text{const}.$ With no logarithm, no subtraction, and no division by a parameter, the standard normal log density is much more efficient to compute than the normal log density with constant location $$0$$ and scale $$1$$.

### 19.1.5 Sampling statement

y ~ std_normal()

Increment target log probability density with std_normal_lupdf(y).
Available since 2.19

### 19.1.6 Stan functions

real std_normal_lpdf(reals y)
The standard normal (location zero, scale one) log probability density of y.
Available since 2.18

real std_normal_lupdf(reals y)
The standard normal (location zero, scale one) log probability density of y dropping constant additive terms.
Available since 2.25

real std_normal_cdf(reals y)
The cumulative standard normal distribution of y; std_normal_cdf will underflow to 0 for $$y$$ below -37.5 and overflow to 1 for $$y$$ above 8.25; the function Phi_approx is more robust in the tails.
Available since 2.21

real std_normal_lcdf(reals y)
The log of the cumulative standard normal distribution of y; std_normal_lcdf will underflow to $$-\infty$$ for $$y$$ below -37.5 and overflow to 0 for $$y$$ above 8.25; log(Phi_approx(...)) is more robust in the tails.
Available since 2.21

real std_normal_lccdf(reals y)
The log of the complementary cumulative standard normal distribution of y; std_normal_lccdf will overflow to 0 for $$y$$ below -37.5 and underflow to $$-\infty$$ for $$y$$ above 8.25; log1m(Phi_approx(...)) is more robust in the tails.
Available since 2.21

R std_normal_qf(T x)
Returns the value of the inverse standard normal cdf $$\Phi^{-1}$$ at the specified quantile x. The std_normal_qf is equivalent to the inv_Phi function.
Available since 2.31

R std_normal_log_qf(T x)
Return the value of the inverse standard normal cdf $$\Phi^{-1}$$ evaluated at the log of the specified quantile x. This function is equivalent to std_normal_qf(exp(x)) but is more numerically stable.
Available since 2.31

real std_normal_rng()
Generate a normal variate with location zero and scale one; may only be used in transformed data and generated quantities blocks.
Available since 2.21