During the process of simulating an exponential random variable

Question

According to the text book, to get a associated CDF $$F_x$$ is given by

$$F_x(a)=\int_{-\infty}^af_x(x)dx=1_{[0, +\infty)}(a)\int_0^a\lambda e^{-\lambda x}dx=1_{[0, +\infty)}(a)(1-e^{-\lambda a})$$

1. It's my first time to see the symbol$$1_{[0, +\infty)}(a)$$ what does that mean?
2. And can you explain that process above step by step?

Answer 1

$$1_{[0, +\infty)}(a) = \begin{cases} 1 & a \in [0, +\infty) \\ 0 & \text{otherwise}\end{cases}$$ In other words, $1_{[0, +\infty)}(a)$ equals $1$ if $a \ge 0$, and otherwise equals $0$.

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$f_X(x)$ equals $\lambda e^{-\lambda x}$ when $x \ge 0$, and equals zero otherwise. So if $a \ge 0$, we have $$\int_{-\infty}^a f_X(x) \, dx = \int_0^a f_X(x) \, dx.$$ Plugging in $\lambda e^{-\lambda x}$ and integrating yields $1-e^{-\lambda x}$.

If instead $a < 0$, then $$\int_{-\infty}^a f_X(x) \, dx = 0$$ because $f_X(x)$ is zero for all $x < 0$.