A calculus exercise from a probability distribution

Several calculus exercises that are derived from a probability problem are presented here.

Let \alpha>0 be a constant.

Evaluate \displaystyle \int_0^1 (1-e^{-\alpha z}) \ dz + \int_1^{\infty}(e^\alpha-1) e^{-\alpha z} \ dz.

Evaluate \displaystyle \int_0^1 z \ (1-e^{-\alpha z}) \ dz + \int_1^{\infty} z \ (e^\alpha-1) e^{-\alpha z} \ dz.

Evaluate \displaystyle \int_0^1 z^2 \ (1-e^{-\alpha z}) \ dz + \int_1^{\infty} z^2 \ (e^\alpha-1) e^{-\alpha z} \ dz.

Background
For anyone who is interested in the origin of these integrals, the first exercise is to verify that the following function is a probability density function (pdf). The second and the third exercises are to find the first and second moment of this probability distribution.

\displaystyle f_Z(z)=\left\{\begin{matrix} \displaystyle 1-e^{-\alpha z}&\ \ \ \ \ \ 0 \le z <1 \\{\text{ }}& \\{\text{ }}& \\{\displaystyle e^{-\alpha z}(e^\alpha-1)}&\ \ \ \ \ \ 1 \le z <\infty  \end{matrix}\right.

The above pdf f_Z(z) is the independent sum of an exponential distribution and the uniform distribution U(0,1). So the answers of the integrals can be derived from knowing these probability distributions. Otherwise, they are excellent exercises in calculus.

\displaystyle \int_0^1 (1-e^{-\alpha z}) \ dz + \int_1^{\infty}(e^\alpha-1) e^{-\alpha z} \ dz=1.

\displaystyle \int_0^1 z \ (1-e^{-\alpha z}) \ dz + \int_1^{\infty} z \ (e^\alpha-1) e^{-\alpha z} \ dz=\frac{1}{2}+\frac{1}{\alpha}.

\displaystyle \int_0^1 z^2 \ (1-e^{-\alpha z}) \ dz + \int_1^{\infty} z^2 \ (e^\alpha-1) e^{-\alpha z} \ dz=\frac{1}{3}+\frac{1}{\alpha}+\frac{2}{\alpha^2}.

Advertisements
This entry was posted in Calculus and tagged , . Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s