y = e^{-|x|}

Graph of the Two-Sided Exponential $y = e^{-|x|}$

$y = e^{-|x|}$ makes the decay of the exponential symmetric about the origin. Also called the two-sided exponential, it becomes the density of the Laplace distribution in probability once multiplied by $\dfrac{1}{2}$ ¹. Set beside the Gaussian $y = e^{-x^2}$ , the differences in the shape of the peak and the weight of the tails are easy to see.

Domain and range

The domain is all real numbers
The range is $0 < y \leq 1$
The maximum $1$ is attained at $x = 0$
It is an even function

Symmetry

Since $f(-x) = f(x)$ the function is even and symmetric about the $y$ -axis. It is two exponential curves joined together: $e^{-x}$ for $x \geq 0$ and $e^{x}$ for $x \leq 0$ .

The corner at the origin

At the origin, where the two pieces meet, the derivatives from the two sides disagree.

Approach	Derivative
from the right	$-1$
from the left	$+1$

The function is therefore not differentiable at the origin, and the graph has a sharp corner. That contrasts with the smooth rounded summit of the Gaussian.

Monotonicity and concavity

For $x > 0$ we have $y' = -e^{-x} < 0$ , so the function decreases, and for $x < 0$ we have $y' = e^{x} > 0$ , so it increases. The second derivative is $y'' = e^{-|x|} > 0$ for $x \neq 0$ , so both sides are concave up and there is no inflection point. As $x \to \pm\infty$ we get $y \to 0$ , making the $x$ -axis a horizontal asymptote.

Comparison with the Gaussian

The two graphs meet at $x = 0$ and $x = \pm 1$ , since $e^{-|x|} = e^{-x^2}$ gives $|x| = x^2$ ; at the crossings the common value is $\dfrac{1}{e} \approx 0.368$ .

$x$	$e^{-\|x\|}$	$e^{-x^2}$
$0.5$	$\approx 0.6065$	$\approx 0.7788$
$1$	$\approx 0.3679$	$\approx 0.3679$
$2$	$\approx 0.1353$	$\approx 0.0183$
$3$	$\approx 0.0498$	$\approx 0.000123$

For $|x| < 1$ the Gaussian is above, and for $|x| > 1$ the two-sided exponential is. In other words the Gaussian bulges more in the middle while the two-sided exponential has the heavier tails. At $x = 3$ the gap is more than four hundredfold.

The Laplace distribution

The integral over the whole line is $2$ , so $\dfrac{1}{2}e^{-|x|}$ is a probability density. That is the standard Laplace distribution.

Quantity	Value
Mean	$0$
Variance	$2$
Mean absolute deviation	$1$

The difference of two independent exponential variables is also known to follow this distribution.

Fourier transform

\int_{-\infty}^{\infty} e^{-|x|}e^{-i\omega x}\,dx = \frac{2}{1+\omega^2}

The image is the shape of the witch of Agnesi. A heavy-tailed peak with a corner is carried over into a smooth rational function.

Applications

In statistics the Laplace distribution is the maximum-entropy distribution when the mean absolute deviation, rather than the variance, is held fixed. It pairs with the fact that the Gaussian is the maximum-entropy distribution when the variance is fixed.

In machine learning, placing a Laplace prior on the coefficients corresponds to $L^1$ regularization. The corner at the origin is what drives many coefficients to be exactly $0$ , giving the sparse solutions that method is known for.

Laplace distribution, Wikipedia

Graph of the Two-Sided Exponential y=e−∣x∣y = e^{-|x|}y=e−∣x∣