Statistics: Could someone show why this exponential pdf integrates into this particular cdf

Solution 1:

For a fixed $\lambda$, let $X$ be the random variable in question. I will denote values of the density and distribution of $X$, simply, as $f(x)$ and $F(x)$, respectively.

By definition, $F(x)=P[X\le x]$. To compute this probability, you would integrate the density, $f$, from $-\infty$ to $x$.

For $x\ge0$:

$$ \eqalign{ F(x) =P[X\le x]&=\int_{-\infty}^x f(t)\,dt\cr &=\int_0^x \lambda e^{-\lambda t}\,dt\cr &= -e^{-\lambda t}\,\bigl|_0^x \cr &=-e^{-\lambda x}-(-e^0)\cr &=1-e^{-\lambda x}. }$$

Note that $f(x)=0$, for $x\le0$; hence the third equality above.

For $x\le0$, when computing $F(x)$, you would be integrating the zero function, and then you would conclude that $F(x)=0$.

Solution 2:

By definition of the cumulative and probability density functions,

$$F(\lambda,x):= \int_{-\infty}^x f(\lambda,u)du$$

Thus, by elementary calculus, for $x\le 0$ we have $F=0$ and for $x>0$,

$$F=\int_{-\infty}^0 0\, du+\int_0^x \lambda e^{-\lambda u}du=[-e^{-\lambda u}]_{0}^{x}=1-e^{-\lambda x}. $$