[Calculus] Exponential Distribution

2021-11-29 10:43 作者:AoiSTZ23 0人讀過 | 我要投稿

By: Tao Steven Zheng (鄭濤)

【Problem】

Calculate the mean, median, variance, and standard deviation for the probability distribution function (PDF)

$p(x)%20%3D%20%5Clambda%20%7Be%7D%5E%7B-%5Clambda%20x%7D%20$

defined on the interval $x%20%5Cin%20%5B0%2C%5Cinfty)$ .

【Solution】

(1) Mean

The mean is the first moment of the PDF.

$%5Cmu%20%3D%20%5Cint_%7B-%5Cinfty%7D%5E%7B%5Cinfty%7D%20x%20%5Ccdot%20p(x)%20dx$

$%5Cmu%20%3D%20%5Cint_%7B0%7D%5E%7B%5Cinfty%7D%20%5Clambda%20x%7Be%7D%5E%7B-%5Clambda%20x%7D%20dx%20$

Using integration by parts yields

$%5Cmu%20%3D%20%5Clim_%7Bb%20%5Cto%20%5Cinfty%20%7D%20%5Cleft%5B-%7Be%7D%5E%7B-%5Clambda%20x%7D%20%5Cleft(x%2B%5Cfrac%7B1%7D%7B%5Clambda%7D%5Cright)%20%5CBiggr%7C_0%5Eb%5Cright%5D%20$

$%5Cmu%20%3D%20%5Cfrac%7B1%7D%7B%5Clambda%7D$

(2) Median

To calculate the median, one must obtain the cumulative distribution function (CDF).

$%20P(x)%20%3D%20%5Cint_%7B0%7D%5E%7Bx%7D%20%5Clambda%20e%5E%7B-%5Clambda%20t%7Ddt$

$P(x)%20%3D%20-%7Be%7D%5E%7B-%5Clambda%20t%7D%20%20%5CBig%7C_0%5Ex$

$P(x)%20%3D%201%20-%20e%5E%7B-%5Clambda%20x%7D%20$

The median is located where the probability is 50%, so

$P(x)%20%3D%201%20-%20e%5E%7B-%5Clambda%20x%7D%20%3D%200.5$

$e%5E%7B-%5Clambda%20x%7D%20%3D%200.5%20$

$x%20%3D%20%5Cfrac%7B%5Cln(0.5)%7D%7B-%5Clambda%7D%20$

$x%20%3D%20%5Cfrac%7B%5Cln(2)%7D%7B%5Clambda%7D%20$

(3) Variance

The variance is calculated by subtracting the square of the mean from the second moment.

$Var(X)%20%3D%20%5Cint_%7B-%5Cinfty%7D%5E%7B%5Cinfty%7D%20x%5E2%20%5Ccdot%20p(x)%20dx%20-%20%7B%5Cmu%7D%5E2$

$Var(X)%20%3D%20%5Cint_%7B0%7D%5E%7B%5Cinfty%7D%20%5Clambda%20x%5E2%20%7Be%7D%5E%7B-%5Clambda%20x%7D%20dx%20-%20%7B%5Cmu%7D%5E2$

Using integration by parts yields (twice), and the mean calculated above yields

$%20Var(X)%20%3D%20%5Clim_%7Bb%20%5Cto%20%5Cinfty%20%7D%20%5Cleft%5B-%7Be%7D%5E%7B-%5Clambda%20x%7D%20%5Cleft(x%5E2%2B%5Cfrac%7B2x%7D%7B%5Clambda%7D%20%2B%20%5Cfrac%7B2%7D%7B%7B%5Clambda%7D%5E2%7D%5Cright)%20%5CBiggr%7C_0%5Eb%5Cright%5D%20-%20%5Cfrac%7B1%7D%7B%7B%5Clambda%7D%5E2%7D$