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Let $X$ be a random variable uniformly distributed on the interval $[0,10]$ and zero elsewhere and let $Y$ be another random variable uniformly distributed on $[0, 20]$ and zero elsewhere. Assuming that $X$ and $Y$ are independent, find

a. $\mathbb{P}(X<4 \cap Y<8)$

b. $\mathbb{E} [X + Y]$

c. $\mathbb{E} [XY]$

d. $\operatorname{Var}(X + Y)$

My work: we will have to double integrate from $0$ to $4$ in $x$ and $0$ to $8$ in $y$, but I am not able to find the function to integrate.

I don't know if this makes sense, please help me with these.

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  • $\begingroup$ @dr.ivanova - The tag you have inserted is incorrect. Probability-theory should not be in this question. $\endgroup$ – Clarinetist Dec 14 '14 at 22:04
  • $\begingroup$ @dr.ivanova, the interval $0<x<10$ is usually denoted by $(0,10)$; the interval $0\le x\le 10$ is usually denoted by $[0,10]$. $\endgroup$ – Joel Reyes Noche Dec 15 '14 at 1:56
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If $X$ is uniformly distributed in $(0, 10)$, then $f_{X}(x) = $ ?.

If $Y$ is uniformly distributed in $(0, 20)$, then $f_{Y}(y) = $ ?.

Since $X$ and $Y$ are independent, then $f_{X, Y}(x, y) = $ ?.

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  • $\begingroup$ fX(x)= 1/10 for 0<x<10 fY(y) 1/20 for 0<y<20 fX,Y(x,y) = fX*fY=1/200 for 0<x,y<10 right? $\endgroup$ – rohit Dec 14 '14 at 21:32
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    $\begingroup$ @rohit: Good. Now integrate that as you stated above in the question. $\endgroup$ – Clarinetist Dec 14 '14 at 21:33
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    $\begingroup$ @rohit - Technically, $f_{X, Y}(x, y) = \dfrac{1}{200}$ for $0 < x < 10$ and $0 < y < 20$. Otherwise it's fine. $\endgroup$ – Clarinetist Dec 14 '14 at 21:34
  • $\begingroup$ ok cool i got it...i got answer for first part as 4/25. Is that right? How about rest of them? $\endgroup$ – rohit Dec 14 '14 at 21:36
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    $\begingroup$ 4/25 is correct. For part b - use the fact that $\mathbb{E}[X+Y] = \mathbb{E}[X] + \mathbb{E}[Y]$. For part c - since $X$ and $Y$ are independent, what do you know about $\mathbb{E}[XY]$? d uses independence as well. $\endgroup$ – Clarinetist Dec 14 '14 at 21:37

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