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Math Help - Probability Distributions

  1. #1
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    Probability Distributions

    Consider a random sample Y1, . . . , Yn from a normal population with mean μ and with variance σ2. Define
    Screen Shot 2012-05-20 at 4.22.25 PM.png
    The question in my textbook then asks:For each of the random variables X1 , . . . , X5 , give its probability distribu-tion. (Do not forget to give the values of the parameters of the distribu-tion.)
    I'm not sure what its asking, could I get some help starting off?
    Attached Thumbnails Attached Thumbnails Probability Distributions-screen-shot-2012-05-20-4.22.25-pm.png  
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  2. #2
    MHF Contributor Siron's Avatar
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    Re: Probability Distributions

    1. X_1=\frac{\overline{Y}-\mu}{\frac{\sigma}{\sqrt{n}}}
    Multiply the numerator and the denominator with a factor n that gives us:
    X_1=\frac{n(\overline{Y}-\mu)}{n\left(\frac{\sigma}{\sqrt{n}}\right)}=\frac  {[Y_1+\ldots+Y_n]-n\mu}{\sqrt{n}\sigma}
    It follows from the central limit theorem now that X_1 has a standard normal distribution.

    2. X_3=\frac{(n-1)S^2}{\sigma^2} has a chi square distribution with n-1 degrees of freedom.

    3. If follows from X_3 that S=\sqrt{\frac{X_3\sigma^2}{n-1}}
    If we substitute this in the epression for X_2 we obtain:
    \frac{\overline{Y}-\mu}{\frac{\sqrt{\frac{X_3\sigma^2}{n-1}}}{\sqrt{n}}}=\frac{X_1}{\sqrt{\frac{X_3}{n-1}}}, with the information that X_1 \sim N(0,1) and X_3 \sim \chi^2_{n-1} if follows that X_2 has a student t distribution with n-1 degrees of freedom.

    4. X_4=X_1^2 and X_1 \sim N(0,1) thus by definition X_4 \sim \chi^2_{1}

    5. X_5=X_1^2+X_3, we know that X_1 \sim N(0,1) and X_3 \sim \chi^2_{n-1} thus  X_3=Z_1^2+\ldots+Z_{n-1}^2 where Z_i \sim N(0,1) (for each i \in \{1,\ldots,n-1\}) therefore X_5 \sim \chi^2_{n}
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