Question

Let x be a continuous random variable over​ [a,b] with probability density function f. Then the median of the​ x-values is that number m for which ModifyingBelow Integral from a to m f (x )font size decreased by 6 dx With font size decreased by 4equalsone half . Find the median. ​f(x)equalsk e Superscript negative kx​, ​[0,infinity​)

Answer 1

Answer:

The answer is "$\int e^{ax}\ dx=\frac{e^{ax}}{a}+c$".

Step-by-step explanation:

Please find the complete question in the attached file.

$\int^{m}_{a}\ f(x)\ dx=\frac{1}{2}\\\\f(x)=ke^{-kx}, \ [0, \infty]$

let m is the median

$so,\\\\ \int^{m}_{a}\ f(x)\ dx=\frac{1}{2}\\\\\to \int^m_0 k\cdot e^{-kx}=\frac{1}{2}\\\\\to [\frac{k\cdot e^{-kx}}{-k}]^m_0=\frac{1}{2}\\\\\to [- e^{-kx}]^m_0=\frac{1}{2}\\\\\to - e^{-km} +1=\frac{1}{2}\\\\\to e^{-km} =\frac{1}{2}\\\\\to e^{km} =2\\\\\to \ln(e^{km}) =\ln 2\\\\\to km= \ln \ 2\\\\\to m=\frac{\ln \ 2}{k}\\\\ \text{the answer is: } \\\\ \to \int e^{ax}\ dx=\frac{e^{ax}}{a}+c$