The exercise
A continuous random variable X, expresses in minutes, the duration of the telephone communications. Its density function is as follows: $f(x)=ke^{-\alpha x}$ if $x>0$, else $f(x)=0$
- a) What conditions must the real $α$ and $k$ verify to be a density function?
If $k=5/6$:
- b) Get the distribution function of X.
- c) Calculates the probability that a communication will last more than three minutes.
- d) Calculates the probability that a communication will last more than five minutes, if it is known to have lasted more than two minutes.
My doubts
- I know what conditions have to meet a density function. But in part (a), how should I analyze these two real numbers? Is it with limits? If so, how should I evaluate them?
- In part (d), should I interpret it as a conditional probability? If so, in what way can I calculate it?
If you have more to say about the rest of the exercise, you are also welcome. Thank you very much.
(a) You want, $f(x)\geq 0$ for all $x$, and $\int_\Bbb R f(x)\operatorname d x=1$. So what constraints on $k, \alpha$ make this possible?
(b) $\mathsf P(X\leq b) = \int_{-\infty}^b f(x)\operatorname d x$
(c) $\mathsf P(X>b) = \int_b^\infty f(x)\operatorname d x$
(d) You want $\mathsf P(X > 5\mid X > 2)$, so use Bayes' Rule and the above.
Tip: $\int_b^c e^{-ax} \operatorname d x = \tfrac 1a(e^{-ab}-e^{-ac})$