What distribution is this?

• Sep 29th 2010, 03:46 AM
noob mathematician
What distribution is this?
Let $\displaystyle s(x)=exp(-(\frac{x}{2})^c)$

$\displaystyle x\ge 0$, $\displaystyle c>0$

How does the distribution look like? Can I conclude that $\displaystyle s(x)$ takes the form of an exponential distribution?
• Sep 29th 2010, 05:06 AM
amul28
yeh...
Quote:

Originally Posted by noob mathematician
Let $\displaystyle s(x)=exp(-(\frac{x}{2})^c)$

$\displaystyle x\ge 0$, $\displaystyle c>0$

How does the distribution look like? Can I conclude that $\displaystyle s(x)$ takes the form of an exponential distribution?

yes this takes form of exponential distribution.
• Sep 29th 2010, 05:18 AM
mr fantastic
Quote:

Originally Posted by noob mathematician
Let $\displaystyle s(x)=exp(-(\frac{x}{2})^c)$

$\displaystyle x\ge 0$, $\displaystyle c>0$

How does the distribution look like? Can I conclude that $\displaystyle s(x)$ takes the form of an exponential distribution?

Since there is no normalising constant (which will be $\displaystyle \frac{1}{\Gamma \left(\frac{c+1}{c}\right)}$, by the way) it does not define a pdf. Overlooking that, I don't think it has any special name. It certainly is NOT an exponential distribution. The closest I can think of is the exponential power distribution (except it's not).
• Sep 29th 2010, 06:44 AM
amul28
O sorry but, y can't it be exponential ?
I thought it is exponential because, as C is a constant, its value can b found by integrating the fn' . Isn't this interpertation correct.
• Sep 29th 2010, 01:25 PM
mr fantastic
Quote:

Originally Posted by amul28
O sorry but, y can't it be exponential ?
I thought it is exponential because, as C is a constant, its value can b found by integrating the fn' . Isn't this interpertation correct.

IF c = 1 then the function, after being multiplied by an appropriate normalising constant, will define an exponential pdf.
• Sep 29th 2010, 10:34 PM
noob mathematician
Actually was asking about how it will look like.. But i guess exponential will be the closer and given that x is only defined >0
• Sep 30th 2010, 03:27 AM
mr fantastic
Quote:

Originally Posted by noob mathematician
Actually was asking about how it will look like.. But i guess exponential will be the closer and given that x is only defined >0

Try different values of c eg. c = 1/2, 1, 2, 3. Plot the graphs. What do you find?

c = 1/2: plot y &#61; Exp&#91;-&#40;x&#47;2&#41;&#94;1&#47;2&#93; - Wolfram|Alpha

c = 1: plot y &#61; Exp&#91;-&#40;x&#47;2&#41;&#94;1&#93; - Wolfram|Alpha

c = 2: plot y &#61; Exp&#91;-&#40;x&#47;2&#41;&#94;2&#93; - Wolfram|Alpha

c = 3: plot y &#61; Exp&#91;-&#40;x&#47;2&#41;&#94;3&#93; - Wolfram|Alpha