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Suppose that $X_1, X_2 \sim U(0,10)$ are continuous random variables. How would I derive the probability density function of $X_1 + X_2$? Clearly we have $$ f_{X_1}(x) = f_{X_2} = 0.1 $$ and so $$ f_{X_1+X_2}(x) = \int_{-\infty}^\infty f_{X_1}(s) \hspace{1mm} f_{X_2}(x-s) \hspace{2mm} ds $$ (if anyone knows what this rule is called, if it has a name, please let me know).

Now, since $f_{X_1}(x) = 0.1$ if $0 \leq x \leq 10$ and is $0$ otherwise, this integral can be simplified to $$ f_{X_1+X_2}(x) = 0.1 \int_0^{10} f_{X_2}(x-s) \hspace{2mm} ds $$

This is where I struggle to see what should happen next. Would anyone be able to help me to understand how to proceed from here?

Cheers.

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  • $\begingroup$ See here: <stats.libretexts.org/Bookshelves/Probability_Theory/…> $\endgroup$
    – JB071098
    Mar 18, 2020 at 15:30
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    $\begingroup$ It sounds like you are talking about "convolution," which is only applicable when $X_1$ and $X_2$ are independent. In your final integral I observe that $f_{X_2}(x-s)$ is only nonzero if $x-s \in [0,10]$, that is, $0\leq x-s \leq 10$. $\endgroup$
    – Michael
    Mar 18, 2020 at 15:52
  • $\begingroup$ What you need to do now is to calculate the integral of a piecewise function - there's no stochastics involved at this point anymore. I.e. define $$ f(x):= f_{X_2}(x) = \begin{cases} 0 &, x<0\\ 0.1 &,x\in [0,10]\\ 0 &, x>10\end{cases}$$ and calculate $\int_0^{10} f(x-s)ds$ $\endgroup$
    – Sudix
    Mar 18, 2020 at 16:42
  • $\begingroup$ Are $X_1$ and $X_2$ independent? $\endgroup$
    – Math1000
    Mar 18, 2020 at 17:57

1 Answer 1

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$Y=X_1+X_2$

$f_Y(y)=0.1 \int_0^{10} f_{X_2}(y-s) ds$

we need to clarify the area $$0<y<20 \, \ 0<s<10 \, \ 0<y-s<10$$

enter image description here

Split in two situation $y<10$ and $10<y<20$ \begin{eqnarray} f_Y(y)&=&\left\{ \begin{array}{cc} 0.1 \int_0^{y} f_{X_2}(y-s) ds & y\leq 10 \\ 0.1 \int_{y-10}^{10} f_{X_2}(y-s) ds & 10<y<20 \\ 0 & O.W \end{array} \right. \\ &=&\left\{ \begin{array}{cc} 0.1 \int_0^{y} 0.1 ds & y\leq 10 \\ 0.1 \int_{y-10}^{10} 0.1 ds & 10<y<20 \\ 0 & O.W \end{array} \right. \end{eqnarray}

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