# Sums of independent r.v

Can someone please explain the last step in this :

It is taken from the book "A First Course In Probability"

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A better way to view the picture : i.stack.imgur.com/VlBMS.png – Belgi Apr 29 '12 at 13:28

By definition, $\color{maroon}{F_X(a)=P[X\le a]=\int_{-\infty}^a f_X(x)\,dx}$.

Now note that in the inner integral on the third line, the $f_Y(y)$ term can be factored out, and thus the integral can be written $$\int_{-\infty}^\infty \int_{-\infty}^{a-y} f_X(x) f_Y(y) \,dx \,dy =\int_{-\infty}^\infty\Biggr[\color{maroon}{ \int_{-\infty}^{a-y} f_X(x) \,dx }\biggl] f_Y(y) \,dy.$$

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Hint

Review the definition of Cumulative distribution function(cdf) of a random variable.

Let $X$ be an absolutely continuous random variable with density $f_X$. Then, the cumulative density function (cdf) is given by:

$$F_X(a)=\int_{-\infty}^af_X(t)\rm{d}t \tag{1}$$

In the text you refer to, A first course in Probability, Sheldon Ross, this is discussed in the introduction to chapter $5$, more precisely in $\S5.1$ $[\text{cf. Pg: 192-193}]$, the equation $(1.2)$ conveys the essence of the equation $(1)$ here.

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