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Math Help - 2 advanced statistics questions

  1. #1
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    2 advanced statistics questions

    Q1:

    A discrete random variable X has expected value μX and standard deviation σX.

    Consider now the random variable Z:

    Z = ( x - μX ) / ( σX )

    What is the expected value and the standard deviation
    of Z?

    Q2:

    Determine the value of the constant A ∈ ℝ such that the function

    g (x) = Ac os x; x ∈ [-π/2 , π/2]; g (x) = 0 x ∉ [π/2-, π/2]

    ...is the density function for a random variable. Also determine the mean and standard deviation of the random variable.


    ____

    Thank you,
    sverige2
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  2. #2
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    Re: 2 advanced statistics questions

    Quote Originally Posted by sverige2 View Post
    Q1:

    A discrete random variable X has expected value μX and standard deviation σX.

    Consider now the random variable Z:

    Z = ( x - μX ) / ( σX )

    What is the expected value and the standard deviation
    of Z?

    Q2:
    Determine the value of the constant A ∈ ℝ such that the function

    g (x) = Ac os x; x ∈ [-π/2 , π/2]; g (x) = 0 x ∉ [π/2-, π/2]

    ...is the density function for a random variable. Also determine the mean and standard deviation of the random variable.
    sverige2
    Q1 is pretty basic.

    $\mu_Z=E[Z]=E\left[\dfrac{X-\mu_X}{\sigma_X}\right]=$

    $\dfrac{E[X]-\mu_X}{\sigma_X}=$

    $\dfrac{\mu_X - \mu_X}{\sigma_X}=0$

    $\mu_Z=0$

    $var[Z]=E[\left(Z-\mu_Z\right)^2]=$

    $var[Z]=E[Z^2]=$

    $E\left[\left(\dfrac{X-\mu_X}{\sigma_X}\right)^2\right]=$

    $\dfrac{1}{\sigma_X^2}E\left[\left(X-\mu_X\right)^2\right]=$

    $\dfrac{1}{\sigma_X^2}var[X]=$

    $\dfrac{1}{\sigma_X^2}\sigma_X^2=1$

    $\sigma_Z=\sqrt{var[Z]}=\sqrt{1}=1$

    Q2

    $\large A\displaystyle{\int_{-\frac n 2}^{\frac n 2}}\cos(x)~dx=1$

    you can do the integration and solve for A

    $\large \mu_G=E[G]=A\displaystyle{\int_{-\frac n 2}^{\frac n 2}}x\cos(x)~dx$

    $\large var[G]=A\displaystyle {\int_{-\frac n 2}^{\frac n 2}}(x-\mu_G)^2\cos(x)~dx$

    $\sigma_G=\sqrt{var[G]}$
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  3. #3
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    Re: 2 advanced statistics questions

    Wow! Thank you so much!

    I have just one curious question... what does E[Z] and E[G] stand for?
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  4. #4
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    Re: 2 advanced statistics questions

    Quote Originally Posted by sverige2 View Post
    Wow! Thank you so much!

    I have just one curious question... what does E[Z] and E[G] stand for?
    the expected values of the random variable Z and of the random variable that has the pdf g(x) (I called it G)
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