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Thread: Independent Variables and Distribution

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    Independent Variables and Distribution




    I am having a lot of trouble starting this problem. Can anyone explain the process in doing this problem. Thank you for any help in advance.

    Suppose that X1, X2 are independent random variables, each of which is uniformly distributed on the integers from 1 to 10. Find the distribution of Y = min(X1, X2).

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    Re: Independent Variables and Distribution


    Start by letting the following:

    U=\min \left ( X1,X2 \right )

    and

    \min \left ( X1,X2) \right )\leq Z.

    As such:

    \Pr \left \{ \min \left ( X1,X2 \right )> Z \right \}= \Pr \left \{ X1> Z \right \}\cap \Pr\left \{ X2> Z \right \}

    \Pr \left \{ U> Z \right \}=\Pr \left \{ X1> Z \right \}\times \Pr \left \{ X2> Z \right \}

    \Pr \left \{ U> Z \right \}=\left ( 1-F_{X1}\left ( Z \right ) \right )\left ( 1-F_{X2}\left ( Z \right ) \right )

    F_{U}\left ( Z \right )=1-\left ( 1-F_{X1} \right )\left ( 1-F_{X2} \right )

    F_{min\left ( X1,X2 \right )}^{Z}=F_{X1}\left ( Z \right )+F_{X2}\left ( Z \right )-F_{X1}\left ( Z \right )\times F_{X2}\left ( Z \right )

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