Weibull & Exponential distribution

**losm1** · June 19th 2010, 02:19 AM

$X$ is a random variable with weibull distribution given with

$ f(x;\alpha ,\beta) = \begin{cases} \frac{\alpha}{\beta^\alpha}x^{\alpha-1}e^{-(x/\beta)^{\alpha}} & x>0\\ 0 & x=0\end{cases} $

Let $\alpha=2$ and $Y=2X^2/\beta^2$ .

Show that $Y$ has exponential distribution and find value of parameter $\lambda$ .

I believe that $G(Y)=P(Y \leq y)$ should be expressed in terms of $X$ , but I don't know where to start.

**SpringFan25** · June 19th 2010, 03:29 AM

you have the right approach

Lets find P(Y<y)
$P(Y \leq y) = P(2x^2/B^2 \leq y) = P(x \leq \sqrt {0.5 B^2 y}) = F(\sqrt {0.5 B^2 y})$

Find (or look up) F(x) then subsititute these values in to get the CDF of Y. Hopefully it will look like an exponential distribution.

**mr fantastic** · June 19th 2010, 03:30 AM

Originally Posted by losm1

$X$ is a random variable with weibull distribution given with

$ f(x;\alpha ,\beta) = \begin{cases} \frac{\alpha}{\beta^\alpha}x^{\alpha-1}e^{-(x/\beta)^{\alpha}} & x>0\\ 0 & x=0\end{cases} $

Let $\alpha=2$ and $Y=2X^2/\beta^2$ .

Show that $Y$ has exponential distribution and find value of parameter $\lambda$ .

I believe that $G(Y)=P(Y \leq y)$ should be expressed in terms of $X$ , but I don't know where to start.

Since you know in advance that the answer is going to be a well-known distribution, a simple approach would be to calculate the moment generating function of Y ....

But if you're determined to use a distribution function approach, then you should start by noting that the cdf of Y is:

$F(y) = \Pr(Y \leq y) = \Pr \left( \frac{2 X^2}{\beta^2} \leq y \right) = \Pr \left( - \frac{\beta}{\sqrt{2}} \sqrt{y} \leq X \leq \frac{\beta}{\sqrt{2}} \sqrt{y} \right)$

$= \Pr \left( 0 < X \leq \frac{\beta}{\sqrt{2}} \sqrt{y} \right)$ since the support of X is $x \geq 0$ .

Now set up the required integral (no need to integrate by the way ....) and then differentiate using the chain rule and the Fundamental Theorem of Calculus.

**losm1** · June 19th 2010, 04:14 AM

Originally Posted by SpringFan25

you have the right approach

Lets find P(Y<y)
$P(Y \leq y) = P(2x^2/B^2 \leq y) = P(x \leq \sqrt {0.5 B^2 y}) = F(\sqrt {0.5 B^2 y})$

Find (or look up) F(x) then subsititute these values in to get the CDF of Y. Hopefully it will look like an exponential distribution.

Thanks, I just plugged in your result in weibull CDF and got exponential CDF with $\lambda=0.5$

I hope that it's correct.

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