The pdf $f(x)$ of an exponential distribution with parameter $\lambda$ is

$$f(x) = \lambda \exp(-\lambda x)\quad x\geq 0$$

The cdf $F(x)$ of such a distribution is

$$F(x) = \int^x_0 f(x') dx'$$

Use your analytical skills to answer these questions:

1. Must the pdfs of two different exponential distributions intersect?

2. In what cases do $F(x)$ and $f(x)$ intersect?

3. Can the cdfs for two exponential distributions intersect?

4. Can the pdfs for three different exponential distibutions intersect at the same point?

You might want to experiment with specific choices of $\lambda$ before constructing more general arguments.

Definite and indefinite integration. Creating and manipulating expressions and formulae. Inequalities. Quadratic equations. Generalising. Mathematical reasoning & proof. Networks/Graph Theory. Probability distributions, expectation and variance. Making and proving conjectures. Probability density functions.