Explaining E[(X−µ)³] calculation and variance formula

[Kior]

I am preparing for my math probability class next semester. There is question: Calculate E(x³_µσ) Would anyone explain the solution in the picture a little bit to me?

1.Why is the step hold? Is there a formula or something that i can calculate E[X-µ]ⁿ?
2.Why is the fourth step hold? Where is the σ from and why variance σ²= E[X-µ]²?

 

[ShayanJ]

If the distribution is continuous, then we have $E[X^n]=\int X^n \rho(X) dX$.
Then it can be proved that $E[aX+c]=aE[X]+c$ which can be used to prove the first formula.
And variance is defined to be the expected value of the squared deviation from the mean, which means $\sigma^2=E[(X-\mu)^2]$.

 

[Ray Vickson]

If the distribution is continuous, then we have $E[X^n]=\int X^n \rho(X) dX$.
Then it can be proved that $E[aX+c]=aE[X]+c$ which can be used to prove the first formula.
And variance is defined to be the expected value of the squared deviation from the mean, which means $\sigma^2=E[(X-\mu)^2]$.

Note that $\sigma^2 \equiv \text{Var}(X)$ is given by
$$\begin{array}{cll} \text{Var}(X)& = E(X - \mu)^2 &\text{definition}\\<br /> & = E(X^2) - \mu^2 & \text{alternative formula}<br /> \end{array}$$
Therefore, we have $E(X^2) = \sigma^2 + \mu^2$.

To get the "alternative formula", just expand out $(X - \mu)^2$ and then take expectations.