Solving Quadratics to Find the Interquartile Range

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Homework Statement
A continous random variable, ##X## has the following probability density function;

##f_{x} =
\begin{cases}
\dfrac{2}{25} (5-x), 0≤t≤5 \\
\\0 , Otherwise
\end{cases}##
Relevant Equations
understanding of probability distribution.
I do not have solution for this; looking forward to your insight.

$$F_{X}=\int_0^m \dfrac{2}{25} (5-x)dx$$
... ending up with
$$\dfrac{2}{25} (5m-\dfrac{m^2}{2})=\dfrac{1}{4}$$ and
$$\dfrac{2}{25} (5m-\dfrac{m^2}{2})=\dfrac{3}{4}$$ we shall end up with two quadratic equations. Solving them gives us;
$$4m^2-40m+25=0$$
$$m=0.669$$ and

$$4m^2-40m+75=0$$
$$m=2.5$$

Therefore our interquartile range is given by;
$$IQR=2.5-0.669=1.83$$ to ##3## decimal places.
 
Last edited:
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I think I would start with \frac{2}{25}\left(5m - \frac{m^2}2\right) = q \in [0,1] and rearrange it into (m-5)^2 = 25(1- q). It is obvious that we want the negative root, so <br /> m = 5(1 - \sqrt{1 - q}). Then the interquartile range is <br /> 5\left(1 - \sqrt{\tfrac14}\right) - 5\left(1 - \sqrt{\tfrac34}\right) = \frac52 \left(\sqrt{3} - 1\right) \approx 1.830.
 
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