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This is a problem I've come across in independent study...

Let $\xi_{1}, \xi_{2}, \xi_{3}$ be i.i.d. N(0,1). Can anything be said about the distribution of $a\max\{\xi_{1}, \xi_{2}\} + b \xi_{3}$ for $a,b>0$? If that is too difficult, it would also help me a lot to know the distribution of $a\max\{\xi_{1}, \xi_{2}\}$ alone. Thanks for all help...

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I assume $a,b \gt 0$ are fixed real numbers, no? – user21436 Apr 17 '12 at 5:33
yes sorry that is unclear. so using your previous hint I have recovered the density of the max to be $2\phi(y)\Phi(y)$... – Red Rover Apr 17 '12 at 5:36
I have edited, I had a blooper there. Now, it should be fine. – user21436 Apr 17 '12 at 5:37
up vote 0 down vote accepted


Let $M=\max\{\xi_1,\xi_2\}$ and $y \in \Bbb R$. Let $\Phi$ represent the distribution of $N(0,1)$.

Let the distribution of $M$ be given by $F_M$. Then,

\begin{align} F_M(y)&=P(M \le y)\\&=P(\max\{\xi_1,\xi_2\} \le y)\\&=P(\xi_1 \le y) \cdot P(\xi_3 \le y)\\&=(\Phi(y))^2 \end{align}

Now, since, $aM$ and $b\xi_3$ are independent, the distribution of the sum should be a routine thing as well.

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