6
$\begingroup$

The probability density function of random variable X is given as $ f_x(x) = \lambda e^ {-\lambda x} , x \ge 0. $ A new random variable $ Y = e^ {-\lambda X}$ is formed. Find the PDF of Y.

  • 0
    @robjohn [See there](http://math.stackexchange.com/q/259098).2012-12-15

2 Answers 2

8

There are shortcuts, but we will use a basic method. The idea is to find the cumulative distribution function of $Y$, and then differentiate to find the density function. We have $F_Y(y)=\Pr(Y\le y)=\Pr(e^{-\lambda X}\le y)=\Pr(-\lambda X \le \log y).$ (We took the logarithm: this preserves inequalities.) Thus $F_Y(y)=\Pr\left(X\ge -\frac{\log y}{\lambda}\right).$ We know that $\Pr(X\ge t)=e^{-\lambda t}$, if $t$ is positive. If $t$ is negative, the probability is $1$.

Substitute for $t$. There is dramatic simplification. The $\lambda$'s cancel, and we get $e^{\log y}$, that is $y$. But note this is correct only when $-\log y$ is $\ge 0$, that is, when $y\le 1$. If $y\gt 1$, $F_Y(y)=1$.

Finally, differentiate. The density function is $1$ on the interval $(0,1)$, and $0$ elsewhere.

  • 0
    Thank you. Its complete and clear.2012-12-11
2

There is a nice "change of variables" formula for doing this. If $g$ is a monotonic function, then

$ f_Y(y)=\left\vert\frac{d}{dy}(g^{-1}(y))\right\vert f_X(g^{-1}(y)) $

So, let $g(x)=e^{-\lambda x}$; this is a monotonic function, so the formula applies. $g^{-1}(y)=-\ln(y)/\lambda$, and so

$ \left\vert\frac{d}{dy}g^{-1}(y)\right\vert=\frac{1}{\lambda y} $

putting the pieces together, we get:

$ f_Y(y)=\frac{1}{\lambda y}\lambda \exp[-\lambda g^{-1}(y)]=\frac{1}{y}\exp[\ln(y)]=1 $

Notice that since $x\geq 0$, $0 (assuming $\lambda>0$) so this distribution makes sense.