What is difference between Exponential distribution function and Cumulative Distribution function

Question

I have two formulas

and

When should we use first and when the second one?

Answer 1

The first is a probability density function; the second a cumulative distribution function.

$$\mathsf P(X\leq x) = \int_{-\infty}^x f_X(s)~\mathrm d ~s\\ 1-e^{-\lambda x} = \int_0^x \lambda e^{-\lambda s}\mathrm d~s\quad \Big[ x>0\Big]$$

The first is obviously most useful in any case you need to know the probability mass of $X$ in some range; though of course you can obtain the result by integration.

The second becomes useful when calculating expectations, and may also be useful in evaluating joint probabilities or convolutions.

Answer 2

Whatever be the law (exponential or other), the cdf is used when you want to connect (and compute) something described in terms of probability of events and something described in terms of (mathematical) analysis. For example:

$$P(X>a \cap X<b)=\int_a^b f_X(x)dx$$

It means that the cdf is frequentely used, sometimes more than the pdf.