The general exercise:

Let's focus on c).

The value of $\sigma$:

$$\sigma =\sqrt{ < x^2 > -< x >^2 }= \frac{1}{\lambda \sqrt{2}}$$

We are asked to plug in $\sigma$:

$$| \Psi (\sigma ) | = \lambda e^{\frac{-2}{\sqrt{2}}}$$

Where $\lambda$ is a positive constant.

I calculated < x > and its value is 0. Check this link to obtain further details:

https://math.stackexchange.com/questions/2893056/misleading-expected-value

**About the probability**

I have difficulties solving this probability as this time I got:

$P_i = \int_{-\sigma}^{\sigma} \lambda e^{-2 \lambda |x|} dx =$

$= \lambda \int_{-\sigma}^{0} e^{-2 \lambda |x|} dx$

$+ \lambda \int_{0}^{+\sigma} e^{-2 \lambda |x|} dx = 0$

Where did I get wrong? I have been trying to get this probability but none outcome seems to make sense.

$$P_o = 1 - P_i $$

Where $P_i$ and $P_o$ mean inside and outside respectively

Thank you

For the probability, you have the right idea but you don't need $\lambda$ in the integral. You are just finding a probability, not an average value within a range. $\lambda$ does not vary anyway so there is no need to integrate to find an average.