Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

If X is Erlang$(k_1,\lambda)$ and Y is Erlang$(k_2,\lambda)$, then is X+Y Erlang$(k_1+k_2,\lambda)$? Do X and Y need to be independent?

share|cite|improve this question
Yes, the distributions are presumed independent in the context of that factoid. –  anon Jul 9 '11 at 7:28
Can you define or give a reference for the distribution "Erlang"? –  Rasmus Jul 9 '11 at 12:45
@Rasmus Wikipedia will do. In summary an erlang$(k,\lambda)$ distribution is a $\Gamma(k,1/\lambda)$. It is sufficient that the erlang variables are independent. I suspect that Jim is asking whether it necessary as well. –  deinst Jul 9 '11 at 15:06

2 Answers 2

If $X$ and $Y$ are independent, that's true. This can be easily seen from the interpretation of an Erlang$(k,\lambda)$ as a sum of $k$ iid exponentials, and also from the characteristic function.

If they are not independent, that's not necesarily true. You can take as a counterexample $X=Y$.

share|cite|improve this answer

No. In general if we have a continuous joint distribution with pdf $f_{X,Y}(x,y)$ and marginals $f_X(x)$ and $f_Y(y)$ and the sum $Z=X+Y\ $ has pdf $$f_Z(z)=\int_{-\infty}^\infty f_X(z-y)f_Y(y) dy$$ that is not enough to be able to say that $X$ and $Y$ are independent. To see this choose two points $(x_1,y_1)$ and $(x_2,y_2)$ and a small value $\delta$ and a value of $\epsilon$ small enough so that the following construction works.

Define $g_{X,Y}(x,y)$ as follows: In a square of side $\delta$ centered on $(x_1,y_1)$ set $g_{X,Y}(x,y)=f_{X,Y}(x,y)+\epsilon$, similarly for a square of side $\delta$ centered on $(x_2,y_2)$, for the squares of side $\delta$ centered on $(x_1,y_2)$ and $(x_2,y_1)$ set $g_{X,Y}(x,y)=f_{X,Y}(x,y)-\epsilon$, and set $g_{X,Y}(x,y)=f_{X,Y}(x,y)$ everywhere else. This leaves the marginals unchanged but disturbs the sum. Now choose some $d$ larger than $\delta$ and let $(x_3,y_3) = (x_1+d,y_1-d)$ and $(x_4,y_4) = (x_2+d,y_2-d)$, and let $h_{X,Y}(x,y)=g_{X,Y}(x,y)-\epsilon$ for squares of side $\delta$ centered on $(x_3,y_3)$ and $(x_4,y_4)$ and $h_{X,Y}(x,y)=g_{X,Y}(x,y)+\epsilon$ on the squares centered on $(x_3,y_4)$ and $(x_4,y_3)$ and $h_{X,Y}(x,y)=g_{X,Y}(x,y)$ elsewhere. Again this has the same marginal distributions, but now the distribution of $X+Y$ is the same for $f_{X,Y}$ and $h_{X,Y}$, and $X$ and $Y$ are not independent in $h_{X,Y}$.

There is probably a simpler construction that shows this.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.