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How would I find the joint pdf of Y1 and Y2, where Y1 = X1 and Y2 is X1+X2+...+Xn, where the Xi's belong to gamma (1, theta)?

This would be a simple exercise is Y2 was X1+X2 or X1+X2+X3, but I don't know how to easily approach this when it is a sum of N random variables.

Thanks!

1 Answers 1

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Assuming the $X_i$ are independent then if you can deal with $Y_2=X_1+X_2+X_3$ then you should be able to deal with the general case.

Each $X_i$ is exponentially distributed with mean $\theta$, so $Y_2-Y_1$ is distributed like a $\Gamma(n-1, \theta)$ distribution.

So the joint density of $Y_1$ and $Y_2$ is $p(y_1,y_2) = \dfrac{(y_2-y_1)^{n - 2}}{(n-2)! \theta^{n}} \exp\left(-\dfrac{y_2}{\theta}\right)$ provided that $ 0 \le y_1 \le y_2$, and $0$ otherwise.