I am asked to calculate the expected value of the maximum of the stochastic process $X(t)=At+B,\,0\le t \le 1$, where both $A$ and $B$ are independent, normally distributed random variables with mean $0$ and variance $\sigma^2$. Any ideas will be much appreciated.
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Well, the maximum is either B or A+B, right? Can you continue? – Did Jan 11 '16 at 14:24
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I don´t get it. Will you please explain? And I need to find the expected value. – RLP Jan 11 '16 at 14:44
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If A=3 and B=2 what is the maximum of At+B over 0<t<1? If A=-2 and B=5 what is the maximum of At+B over 0<t<1? – Did Jan 11 '16 at 15:07
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For each $t$ the variables $A$ and $B$ take different values. The question really is $Max X(t)$ over the interval $[0,1]$ – RLP Jan 11 '16 at 15:26
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"For each t the variables A and B take different values." What? Are you saying that $X(t)=A(t)t+B(t)$ for some i.i.d. random variables $(A(t))$ and $(B(t))$? – Did Jan 11 '16 at 15:28
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I may not be reading the problem correctlly. Assuming I am wrong, how do I find $E[MaxX(t)]$? – RLP Jan 11 '16 at 15:48
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Thanks a lot, @Did. You were very kind and helpful. – RLP Jan 11 '16 at 22:21