Fundamentals of Probability Theory - 12 - Laplace Distribution

This article documents the Laplace distribution.

Laplace distribution

• Probability Density Function:

p(x | \mu, \gamma)=\frac{1}{2 \gamma} \exp \left(-\frac{|x-\mu|}{\gamma}\right)

The density function of the Laplace distribution can be viewed as a "back-to-back" splicing of the probability densities of two exponential distribution functions.

• Expect:

\quad \mathbb{E}[X]=\mu

• Variance:

\operatorname{Var}[X]=2 \gamma^{2}

Laplace distribution and positiveState distribution

The probability density of the Laplace distribution looks very similar to the normal distribution, but it is a bit sharper (concentrated) than the normal distribution

The 0.99th percentile of the standard Laplace distribution is 3.91, while the standard normal distribution is 2.32, which means that the random variable obeying the Laplace distribution, the probability of an extremely large value, must bemuch larger than the normal distribution.

Parameters of Laplace distributionEstimate

• The sample median of the Laplace distribution is the maximum likelihood estimate of the parameter \mu

\quad \hat{b}=\frac{1}{N} \sum_{i=1}^{N}\left|x_{i}-\hat{\mu}\right|

References

• http://www.huaxiaozhuan.com/Fundamentals of Mathematics/chapters/2_probability.html
• https://zhuanlan.zhihu.com/p/156234503