# Can a two parameter Weibull Distribution be written as an exponential family form?

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Keqiao Li on 27 Mar 2017
Commented: Keqiao Li on 28 Mar 2017
Hi guys, I was wondering whether the two parameter Weibull Distribution belongs to a exponential family?
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John BG on 27 Mar 2017
Hi Keqiao
Weibull with k=1 is exponential . .

John BG on 27 Mar 2017
Keqiao
makedist generates exactly the same points when the setting
Weibull a = Exponential mu
a:scale b:shape
.
pd1=makedist('Weibull','a',2,'b',2) % a:scale b:shape
nx=[0:.1:10]
y=pdf(pd1,nx)
plot(y,'r');grid on
hold all
pd2=makedist('Weibull','a',2,'b',1) % a:scale b:shape
y2=pdf(pd2,nx)
plot(y2,'b');
pd3=makedist('Exponential','mu',2) % a:scale b:shape
y3=pdf(pd3,nx)
plot(y3,'k')
. .
y2 blue and y3 black are completely overlapped.
isequal(y2,y3)
ans =
1
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thanks in advance for time and attention
John BG
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Keqiao Li on 28 Mar 2017
Hi John, thank you for your answer. Your answer is quite useful. But since Weibull distribution can be reduced to exponential distribution, this is because the shape parameter is "1" and a = mu? Is this the only condition when Weibull distribution can be considered as a exponential family (not just exponential distribution)?