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Math Help - Order statistic

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    Order statistic

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    Re: Order statistic

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    Re: Order statistic

    Quote Originally Posted by variousbubble
    Suppose that X_1, \ldots, X_5 are i.i.d. uniform random variables on the interval (3,6). Let \bar{X} be the sample mean, with mean \mu_{\bar{X}} and standard deviation \sigma_{\bar{X}}. Find the probability that the first order statistic Y_1 and the fifth order statistic Y_5 differ from \mu_{\bar{X}} by less than \sigma_{\bar{X}}.

    f_X(x)=\frac{1}{b-a}=\frac{1}{6-3} = \frac{1}{3} , \hspace{0.5cm} 3<x<6

    F_X(x)=\int^{t=x}_{t=3}\frac{1}{3} dt = \frac{t}{3}\vert^{t=x}_{t=3} =  \frac{x}{3}-1 , \hspace{0.5cm} 3<x<6

    \mu = \frac{a+b}{2} = \frac{3+6}{2} = \frac{9}{2} = 4.5

    \sigma^{\color{red}{2}\color{black}} = \frac{(b-a)^2}{12} = \frac{(6-3)^2}{12} = \frac{9}{12} = 0.75 \implies \sigma=\sqrt{0.75}

    \mu_{\bar{X}} = \mu = 4.5

    \sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}} = \frac{\sqrt{0.75}}{\sqrt{5}} = \sqrt{0.15} \approx 0.3873

    P\left(\left[|Y_1-\mu_{\bar{X}}| < \sigma_{\bar{X}}\right] \cap \left[|Y_5-\mu_{\bar{X}}| < \sigma_{\bar{X}}\right] \right) \approx P\left(\left[4.1127 < Y_1 < 4.8873\right] \cap \left[4.1127 < Y_5 < 4.8873\right] \right)

    If both the sample min and sample max of the X_is are bounded by 4.1127 and 4.8873, then all X_is from the sample are bounded by 4.1127 and 4.8873.

    Thus,

    P\left(\left[4.1127 < Y_1 < 4.8873\right] \cap \left[4.1127 < Y_5 < 4.8873\right] \right) = P\left(\left[4.1127 < X_1 < 4.8873\right] \cap \left[4.1127 < X_2 < 4.8873\right] \cap \cdots \cap \left[4.1127 < X_5 < 4.8873\right]\right)

    Since the X_is are independent, we can rewrite the right-hand side from above as

    P\left(4.1127 < X_1 < 4.8873\right) \cdot P\left(4.1127 < X_2 < 4.8873\right) \cdot \cdots \cdot P\left(4.1127 < X_5 < 4.8873\right) ,

    or, equivalently,

    \left[P\left(4.1127 < X_1 < 4.8873\right)\right]^5,

    which is,

    \tiny{\left[F_X(4.8873) - F_X(4.1127)\right]^5 = \left[\left(\frac{4.8873}{3}-1\right) - \left(\frac{4.1127}{3}-1\right)\right]^5 \approx 0.2582^5 \approx 0.00115}.
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