Given that a variable π is uniformly distributed at point (2,5). Find a) the density function of π. b) π ( 5/2 β€ π β€ 4).
We have been given that \(Y\) is uniformly distributed at point \((2,5)\), i.e., we have \(Y \sim U(2,5)\)
a) \(1 / 3\)
The density function of \(Y\) which follows Uniform \((a=2, b-5)\) is given as
$$ f(y)=\frac{1}{b-a}=\frac{1}{5-2}=\frac{1}{3} $$
b) \(0.5\)
We know that the PDF of \(Y\) is given as
$$ \begin{aligned} &P(Y \leq y) \\ &=\frac{y-2}{5-2}=\frac{y-2}{3} \end{aligned} $$
We need
$$ \begin{aligned} &P\left(\frac{5}{2} \leq Y \leq 4\right) \\ &=P(Y \leq 4)-P\left(Y \leq \frac{5}{2}\right) \\ &=\frac{4-2}{3}-\frac{\frac{5}{2}-2}{3} \\ &=\frac{2}{3}-\frac{1}{6} \\ &=0.5 \end{aligned} $$
Hence
$$ P\left(\frac{5}{2} \leq Y \leq 4\right)=0.5 $$
Given that a variable π is uniformly distributed at point (2,5). Find a) density function of Y b) π ( 5/2 β€ π β€ 4).
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