# Would anyone explain the solution a little bit to me?

• Kior
In summary, the conversation discusses the calculation of E(x3µσ) and the formula for calculating E[X-µ]n. It also explains the steps involved in the calculation and the concept of variance. The alternative formula for variance is also mentioned.
Kior
Member warned about posting without the template and with no effort
I am preparing for my math probability class next semester. There is question: Calculate E(x3µσ) 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?

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] ##.

Shyan said:
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
$$\begin{array}{cll} \text{Var}(X)& = E(X - \mu)^2 &\text{definition}\\ & = E(X^2) - \mu^2 & \text{alternative formula} \end{array}$$
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.

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