1. ## 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( . )

2. 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( . )
$F_Z (z) = P(Z \leq z) = \{ {\begin{array}{ll} P(\alpha Z \leq \alpha z) & if \alpha > 0 \\ P(\alpha Z \geq \alpha z) = 1 - P(\alpha Z \leq \alpha z) & if \alpha < 0\\ \end{array}}$

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

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

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

3. thanks alot kalagota for this info