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  1. #1
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    cumulative

    suppose that X has acummulative distribution function (CDF) FX(x) and that Z=alpha.X+beta, where alpha and beta are two constants such that alpha not equal zero. find CDF Fz(ζ) in term of Fx( . )
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by compufatwa View Post
    suppose that X has acummulative distribution function (CDF) FX(x) and that Z=alpha.X+beta, where alpha and beta are two constants such that alpha not equal zero. find CDF Fz(ζ) in term of Fx( . )
    F_Z (z) = P(Z \leq z) = \{ {\begin{array}{ll} P(\alpha Z \leq \alpha z) & if \alpha > 0 \\ P(\alpha Z \geq \alpha z) = 1 - P(\alpha Z \leq \alpha z) & if \alpha < 0\\ \end{array}}

     = \{ {\begin{array}{ll} P(\alpha Z + b \leq \alpha z + b) & if \alpha > 0 \\ P(\alpha Z + b \geq \alpha z + b) = 1 - P(\alpha Z + b \leq \alpha z + b) & if \alpha < 0\\ \end{array}}

     = \{ {\begin{array}{ll} P(X \leq \alpha z + b) & if \alpha > 0 \\ P(X \geq \alpha z + b) = 1 - P(X \leq \alpha z + b) & if \alpha < 0\\ \end{array}}

     = \{ {\begin{array}{ll} F_X(\alpha z + b) & if \alpha > 0 \\ P(X \geq \alpha z + b) = 1 - F_X(\alpha z + b) & if \alpha < 0\\ \end{array}}
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  3. #3
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    thanks alot kalagota for this info
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