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I am solving a problem where $X$ is an exponential random variable and $\lambda=\frac{1}{10}$. I need to find the CDF of $X$ and have that $\int_0^\infty \frac{e^\frac{-x}{10}}{10}$ turns out to be $1$, however the answer is $1-e^\frac{-x}{10}$. Where did I go wrong?

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The cumulative distribution function, $F_X$, is a function; its value at the point $a\ge 0$ is $F_X(a)=P[X\le a]=\int_0^a{\textstyle {1\over10}} e^{-x/10}\,dx.$ Note, also, that $F_X(a)=0$ for $a<0$.

Of course, use $x$ if you like for the independent variable.

After doing the integration, you should find: $ F_X(x)=\cases{1-e^{-x/10},&$x\ge0$\cr 0,&$x<0$ }. $

What you found was "$F(\infty)$", which is of course 1.

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    While it is for$a$different pdf, you might want to read [this](http://math.stackexchange.com/a/118757/15941) and the comments thereafter.2012-03-20