The image shows the calculation of the expectation value $E(x^2)$, which is typically done for a normal distribution, where the probability density function (PDF) is given by $f(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2 / 2}$. Let's go through each step:

1. First Line: $E(x^2) = \int_{-\infty}^{\infty} x^2 f(x) \, dx$

   • This is the definition of the expected value of $x^2$, where $f(x)$ is the probability density function (presumably for a normal distribution).

2. Second Line: $E(x^2) = \int_{-\infty}^{\infty} x^2 e^{-x^2 / 2} \, dx$

   • The PDF $f(x)$ is replaced with $e^{-x^2 / 2}$, which implies a standard normal distribution (ignoring constants).

3. Third Line: A substitution is made: $u = \frac{x^2}{2}$, which gives $x^2 = 2u$ and $du = x dx$.

   • The integral is transformed using this substitution, and the bounds change accordingly. This leads to: $\int_0^\infty (2u) e^{-u} du$

4. Fourth Line: Simplifies to: $4 \int_0^\infty u e^{-u} du$ This is a standard Gamma function integral $\Gamma(n) = \int_0^\infty u^{n-1} e^{-u} du$, with $n = 2$.

5. Final Line: The result of the Gamma function $\Gamma(2) = 1! = 1$, so the integral becomes: $4 \times 1 = 4$

Conclusion:

The steps are correct, and the result $E(x^2) = 4$ is true, assuming the constants associated with the normal distribution (like $\frac{1}{\sqrt{2\pi}}$) are omitted in the setup.

Let me know if you'd like more details on any of these steps!

Here are some related questions to expand your understanding:

1. What is the role of the Gamma function in solving integrals?
2. How would the result change if the normalization constant $\frac{1}{\sqrt{2\pi}}$ was included?
3. Can you explain the method of substitution used in this problem?
4. What is the expected value of $x^2$ for a non-standard normal distribution (with mean and variance)?
5. How does this approach differ when calculating the variance $\text{Var}(x) = E(x^2) - (E(x))^2$?

Tip: When performing substitutions in integrals, always remember to change the limits of integration to match the new variable of substitution.