Based the density function of the observed outcome, the probability [tex]P(x\leq \frac{1}{3} )[/tex] is equal to 0.1388.
A density function can be defined as a type of function which is used to represent the density of a continuous random variable that lies within a specific range.
Given the following density function:
[tex]f(x)=\left \{ {{2(1-x),\;0 < x < 1} \atop {0,\; otherwise}} \right.[/tex]
Next, we would calculate [tex]P(x\leq \frac{1}{3} )[/tex];
[tex]P(x\leq \frac{1}{3} )=\int\limits^\frac{1}{3} _0 {f(x)} \, dx \\\\P(x\leq \frac{1}{3} )=2 \int\limits^\frac{1}{3} _0 {(1-x)} \, dx \\\\P(x\leq \frac{1}{3} )=2|x-\frac{x^2}{2} |\limits^\frac{1}{3} _0\\\\P(x\leq \frac{1}{3} )=2[\frac{1}{3} -\frac{1}{2} (\frac{1}{3})^2]\\\\P(x\leq \frac{1}{3} )=2[\frac{1}{3} -\frac{1}{18}]\\\\P(x\leq \frac{1}{3} )=0.1388[/tex]
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