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

Hi guys, I was wondering whether the two parameter Weibull Distribution belongs to a exponential family?

[John BG]

Hi Keqiao

Weibull with k=1 is exponential .

.

Answer 1

[John BG]

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]

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)?