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

The answer is "
`\int e^(ax)\ dx=(e^(ax))/(a)+c`".

Explanation:

Please find the complete question in the attached file.

`\int^(m)_(a)\ f(x)\ dx=(1)/(2)\\\\f(x)=ke^(-kx), \ [0, \infty]\\\\`

let m is the median

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