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Suppose that ##Y=\sum_{k=1}^KX_{(k)}##, where ##X_{(1)}\leq X_{(2)}\leq\cdots X_{(N)}## and (##N\geq K##). I want to find the characteristic function of ##Y## as

[tex]\phi(jvY)=E\left[e^{jvY}\right]=E\left[e^{jv\sum_{k=1}^KX_{(k)}}\right][/tex]

In the case where ##\{X\}## are i.i.d random variables, the above characteristic function will be

[tex]\phi(jvY)=\prod_{k=1}^K\phi(jvX_k)[/tex]

but when the random variables are ordered, they are no longer independent. How can the characteristic function be found in this case?

[tex]\phi(jvY)=E\left[e^{jvY}\right]=E\left[e^{jv\sum_{k=1}^KX_{(k)}}\right][/tex]

In the case where ##\{X\}## are i.i.d random variables, the above characteristic function will be

[tex]\phi(jvY)=\prod_{k=1}^K\phi(jvX_k)[/tex]

but when the random variables are ordered, they are no longer independent. How can the characteristic function be found in this case?

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