Question

exponential distribution

x1 has density f1(x)=e^-x with r.v if Xa=X1/a then it has density fa(x)=ae^-ax

can someone tell me what happened here???? dont get why will become the parameter is a but not 1/a since we divide the r.v by a

Answer 1

f(x)= e^-x, x>0..

Xa=x/a

Or x=Xa ×a, if we find jacobin of this then d(x)/dXa =a ;{d /dXa represent differencial function}

Now, to find density of transformed variable, we need to multiply the density of x by its jacobin and replace in its density x by transformed variable Xa. So

f(Xa)= f(x)×modulus(dx/dXa) = e^-(a×Xa)×modulus(dx/dXa) =e^-(Xa×a) .a =ae^(-aXa). This is the density of exponential distribution with parameter a.

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