cumulative

**compufatwa** · November 15th 2007, 08:40 AM

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( . )

**kalagota** · November 15th 2007, 02:58 PM

Originally Posted by compufatwa

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( . )

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

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

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

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

**compufatwa** · November 16th 2007, 04:40 AM

thanks alot kalagota for this info

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