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Would anyone explain the solution a little bit to me?

  1. Jun 14, 2015 #1
    • 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?
    media%2Ff8e%2Ff8e1f936-5fdb-445c-8fef-dd01be6d4e86%2Fphp7cYPNz.png
     
  2. jcsd
  3. Jun 14, 2015 #2

    ShayanJ

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    Gold Member

    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] ##.
     
  4. Jun 14, 2015 #3

    Ray Vickson

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    Homework Helper

    Note that ##\sigma^2 \equiv \text{Var}(X)## is given by
    [tex] \begin{array}{cll} \text{Var}(X)& = E(X - \mu)^2 &\text{definition}\\
    & = E(X^2) - \mu^2 & \text{alternative formula}
    \end{array}[/tex]
    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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