$Foundations in probability$

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$Foundations in probability$

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Probability

Burgoos

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Matchmaticians Foundations in probability File #1

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Burgoos

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philip can you check my other question as well, its similar but my deadline is wrong
Erdos

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I will.

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Burgoos

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philip can you check my other question as well, its similar but my deadline is wrong

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We have
\[f_Y(y)=\int_{0}^{\infty}\frac{1}{2}ye^{-xy}dx=-\frac{1}{2}e^{-xy}|_{x=0}^{x=\infty}=0+\frac{1}{2}=\frac{1}{2}.\]
Then
\[f(x|y)=\frac{f(x,y)}{f_Y(y)}=\frac{\frac{1}{2}ye^{-xy}}{\frac{1}{2}}=ye^{-xy}.\]
So
\[E[e^{X/2}|Y=1]=\int_0^{\infty}e^{x/2}f(x|y=1)dx=\int_0^{\infty}e^{x/2}(1e^{-x\cdot 1})dx=\int_0^{\infty}e^{-x/2}dx\]
\[=-2e^{-x/2}|_0^{\infty}=0-(-2)=2.\]

Erdos

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$Foundations in probability$

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