Consider the following function where fx and Fx are the pdf

Consider the following function:

where f(x) and F(x) are the pdf and cdf of X respectively. If X is exponentiallydistributed, what is r(x)?

Solution

Given that the function,

r(x) = f(x) / [ 1-F(x) ]

where f(x) is pdf and F(x) is cdf of X.

Also given that X is exponentially distributed then the pdf of X is,

f(x) = e^- x x 0 and >0

where   is the parameter of exponential distribution.

We find first cdf of X,

cdf [F(x)] = f(x) dx (range is from 0 to x)

=   e^- x dx   (range is from 0 to x)

=   e^- x dx    (range is from 0 to x)

=   [ e^- x / - ]   (range is from 0 to x)

= - [ e^- x - e⁰ ]   (range is from 0 to x)

= - [ e^- x - 1 ]   

F(x) = 1 - e^- x    x 0 and >0

r(x) = [ e^- x ] / { 1 - [ 1 - e^- x ] }

= e^- x / e^- x

r(x) =