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I'm having a hard time proving the following proposition:

Let $X_1,X_2,dots,X_n$ be i.i.d exponentially distributed random variables with parameter $lambda$. Then the joint distribution of $X_1, X_1+X_2, X_1+X_2+X_3,dots,X_1+dots+X_n-1$ given the condition $X_1+dots+X_n=t$ is equal to the joint distribution of the order statistic $(Y_1,dots,Y_n-1)$ from the uniform distribution on $[0,t]$.

So $f(x_1,dots,x_1+dots+x_n-1 |x_1+dots+x_n=t)=dfracf(x_1,dots,x_1+dots+x_n)f(x_1+dots+x_n)$. And I think now I'm supposed to use the memorylessness property of the exponential distribution but I don't see how.

Edit: I tried to transform the variables and use the Jacobi determinant but I'm still stuck:

probability statistics

share|cite|improve this question

edited Mar 17 at 10:50

liz

asked Mar 16 at 8:56

lizliz

1047

$endgroup$

add a comment | 

0

$begingroup$

I'm having a hard time proving the following proposition:

Let $X_1,X_2,dots,X_n$ be i.i.d exponentially distributed random variables with parameter $lambda$. Then the joint distribution of $X_1, X_1+X_2, X_1+X_2+X_3,dots,X_1+dots+X_n-1$ given the condition $X_1+dots+X_n=t$ is equal to the joint distribution of the order statistic $(Y_1,dots,Y_n-1)$ from the uniform distribution on $[0,t]$.

So $f(x_1,dots,x_1+dots+x_n-1 |x_1+dots+x_n=t)=dfracf(x_1,dots,x_1+dots+x_n)f(x_1+dots+x_n)$. And I think now I'm supposed to use the memorylessness property of the exponential distribution but I don't see how.

Edit: I tried to transform the variables and use the Jacobi determinant but I'm still stuck:

probability statistics

share|cite|improve this question

edited Mar 17 at 10:50

liz

asked Mar 16 at 8:56

lizliz

1047

$endgroup$

add a comment | 

$begingroup$

I'm having a hard time proving the following proposition:

Let $X_1,X_2,dots,X_n$ be i.i.d exponentially distributed random variables with parameter $lambda$. Then the joint distribution of $X_1, X_1+X_2, X_1+X_2+X_3,dots,X_1+dots+X_n-1$ given the condition $X_1+dots+X_n=t$ is equal to the joint distribution of the order statistic $(Y_1,dots,Y_n-1)$ from the uniform distribution on $[0,t]$.

So $f(x_1,dots,x_1+dots+x_n-1 |x_1+dots+x_n=t)=dfracf(x_1,dots,x_1+dots+x_n)f(x_1+dots+x_n)$. And I think now I'm supposed to use the memorylessness property of the exponential distribution but I don't see how.

Edit: I tried to transform the variables and use the Jacobi determinant but I'm still stuck:

probability statistics

share|cite|improve this question

edited Mar 17 at 10:50

liz

asked Mar 16 at 8:56

lizliz

1047

$endgroup$

share|cite|improve this question

edited Mar 17 at 10:50

liz

asked Mar 16 at 8:56

lizliz

1047

share|cite|improve this question

edited Mar 17 at 10:50

liz

asked Mar 16 at 8:56

lizliz

1047

asked Mar 16 at 8:56

lizliz

1047

asked Mar 16 at 8:56

lizliz

1047