91Ó°ÊÓ

The random variable X follows gamma distribution with parametersα and β,and c is a positive constant.

Let,\(f\left( x \right)\)denote the p.d.f of X, then according to definition of gamma distribution,

\(f\left( x \right) = \frac{{{\beta ^\alpha }}}{{\Gamma \left( \alpha \right)}}{x^{\alpha - 1}}{e^{ - \beta x}}\;\;\;for\;x > 0\)\(\)

Now,

Let\(Y = cX\).

Then,\(X = \frac{Y}{c}\).

Since,\(dx = \frac{{dy}}{c}\), then for\(x > 0\),

\(\begin{aligned}{c}g\left( y \right) &= \frac{1}{c}f\left( {\frac{y}{c}} \right)\\ &= \frac{1}{c}\frac{{{\beta ^\alpha }}}{{\Gamma \left( \alpha \right)}}{\left( {\frac{y}{c}} \right)^{\alpha - 1}}{e^{ - \beta \left( {\frac{y}{c}} \right)}}\end{aligned}\)

Therefore,

\(g\left( y \right) = \frac{{{{\left( {\frac{\beta }{c}} \right)}^\alpha }}}{{\Gamma \left( \alpha \right)}}{y^{\alpha - 1}}{e^{ - \left( {\frac{\beta }{c}} \right)y}}\)………………….. (1)

Hence equation (1) shows that Y follows gamma distribution with parameters α and \(\beta /c\).