Given probability density function of a speed of a car $P(s) = Ase^{\frac{-s}{s_0}}$, what's the mean value of the speed?

Question

$A \in \mathbb{R}^+$; $s_o \in \mathbb{R}^+$; $s \in \mathbb{R}^+ \cup {\{0}\}$; $s$ - speed of a car.

Definiton of mean: $$\displaystyle<s> = \int_{-\infty}^{+\infty} sP(s)ds,$$ but in this case wouldn't it be

$$\displaystyle<s> = \int_{-\infty}^{+\infty} sP(s)ds = A\int_{-\infty}^{+\infty} s^2e^{\frac{-s}{s_0}}ds = \infty$$ ?

Clearly I don't understand something; any help will be appreciated.

edit

Can someone point out the mistake(s)? $$\displaystyle<s> = \int_{-\infty}^{+\infty} sP(s)ds = A\int_{-\infty}^{+\infty} s^2e^{\frac{-s}{s_0}}ds.$$ Let $$u = \frac{-s}{s_0} \Longrightarrow ds = -s_0du, \\ s = -\infty \Longrightarrow u = +\infty, \\ s = +\infty \Longrightarrow u = -\infty,$$ Then $$<s> = As_0^3\int_{-\infty}^{\infty} u^2e^udu = As_0^3 \bigg[ u^2e^u \bigg\rvert_{-\infty}^{+\infty} - 2\int_{-\infty}^{\infty} ue^udu\bigg] = As_0^3 \bigg[ u^2e^u \bigg\rvert_{-\infty}^{+\infty} - 2\bigg(ue^u\bigg\rvert_{-\infty}^{+\infty}-\int_{-\infty}^{\infty} e^udu\bigg)\bigg] = As_0^3 \bigg[ u^2e^u \bigg\rvert_{-\infty}^{+\infty} - 2ue^u\bigg\rvert_{-\infty}^{+\infty}+2e^u\bigg\rvert_{-\infty}^{+\infty}\bigg] =^? As_0^3\bigg[(\infty - 0)- (\infty - 0) + (\infty - 0)\bigg] = \infty, $$ i.e. it diverges.

edit

Ok, so the integral should be from $0$ to $\infty$, since $s \ge 0$, and such integral nicely converges to $$2As_0^3.$$

Answer 1

The mistake in your computation is not in the calculation, which is correct - that integral is divergent. The mistake is that $P(s) = 0$ for $s < 0$, and so the correct integral is

$$A \int_{0}^{\infty} s^2 e^{-s/s_0} \, ds$$

which you should easily find is convergent.

Answer 2

You're integral looks correct, assuming A is a constant. But the result of the integral does not equal $\infty$. You'll have to work through that integral (looks like integration by parts will help you here) and then get a result.

Edit: just saw that you worked through the integral and still got it to equal $\infty$. Are you sure that your bounds on the integral are correct? Is the lower bound supposed to be $0$ instead of $-\infty$ since that car can't go negative speed?