Question

If \(X\) has a Pareto distribution with parameters \(\alpha\) and \(\beta\) and if \(c\) is a positive constant, show that \(Y=c X\) has a Pareto distribution with parameters \(\alpha\) and \(\beta / c\)

Short answer

The transformed variable \(Y = cX\) has a Pareto distribution with parameters \(\alpha\) and \(\beta / c\).

Step-by-step solution

Definition of Pareto Distribution

Recall the cumulative density function (CDF) for a Pareto-distributed random variable \(X\) with parameters \(\alpha\) and \(\beta\) is \(F_{X}(x) = 1 - (\beta / x)^{\alpha}\) for \(x \geq \beta\).

Transform the Random Variable

Define a new random variable \(Y = cX\) where \(c\) is a positive constant. The cumulative density function (CDF) of \(Y\) can be written using \(F_{Y}(y) = P(Y \leq y) = P(cX \leq y) = P(X \leq y/c) = F_{X}(y/c)\).

Substitute X CDF into Y CDF

Now, substituting the CDF of \(X\) from step 1 into the expression for \(F_{Y}\) from step 2, we get \(F_{Y}(y) = 1 - (\beta / (y/c))^{\alpha} = 1 - ((\beta / c) / y)^{\alpha}\). This is the cumulative density function for a Pareto distribution with parameters \(\alpha\) and \(\beta / c\).

Conclusion

Thus, a variable \(Y = cX\), where \(X\) is a Pareto-distributed variable with parameters \(\alpha\) and \(\beta\) and \(c\) is a positive constant, has a Pareto distribution with parameters \(\alpha\) and \(\beta / c\).