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Chapter 4: Q11E (page 234)

Let X have the uniform distribution on the interval\(\left( {0,1} \right)\). Find the IQR of X.

Short Answer

Expert verified

The IQR of X is 0.5.

Step by step solution

01

Given information

Xfollows the uniform distribution on the interval \(\left( {0,1} \right)\).

02

Find the inter-quartile range of X

\(X \sim Uniform\left( {0,1} \right)\)

So, \(\begin{aligned}{}F\left( x \right) &= \frac{{x - 0}}{{1 - 0}}\\ &= x;0 < x < 1\end{aligned}\)

Let \(F\left( {{x_1}} \right) = 0.25\) and \(F\left( {{x_3}} \right) = 0.75\).

Using the CDF form of X, it is observed that \({x_1} = 0.25\) and \({x_3} = 0.75\).

The interquartile range of X is derived as shown below.

\(\begin{aligned}{}{F^{ - 1}}\left( {0.75} \right) - {F^{ - 1}}\left( {0.25} \right) &= {x_3} - {x_1}\\ = 0.75 - 0.25\\ &= 0.5\end{aligned}\)

Therefore, the IQR of X is 0.5.

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