91Ó°ÊÓ

Xis a random variable whose interquartile range is \(\eta \), and Y is another random variable, where\(Y = 2X\).

The interquartile range of X can be mathematically presented as \(F_X^{ - 1}\left( {0.75} \right) - F_X^{ - 1}\left( {0.25} \right) = \eta \).

Now,

\(\begin{aligned}{}{F_Y}\left( y \right) &= P\left( {Y \le y} \right)\\ &= P\left( {2X \le y} \right)\\ &= P\left( {X \le \frac{y}{2}} \right)\\ &= {F_X}\left( {\frac{y}{2}} \right)\end{aligned}\)

Thus, \(F_X^{ - 1}\left( a \right) = \frac{1}{2}F_Y^{ - 1}\left( a \right)\).

Again,

\(\begin{aligned}{}F_X^{ - 1}\left( {0.75} \right) - F_X^{ - 1}\left( {0.25} \right) &= \eta \\\frac{1}{2}F_Y^{ - 1}\left( {0.75} \right) - \frac{1}{2}F_Y^{ - 1}\left( {0.25} \right) = \eta \\F_Y^{ - 1}\left( {0.75} \right) - F_Y^{ - 1}\left( {0.25} \right) = 2\eta \end{aligned}\)

Thus, the IQR of Y is \(2\eta \).