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April 26th, 2017, 10:25 AM   #1
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Finding a cdf

2 Choose a number U from the interval [0, 1] with uniform distribution. Find
the cumulative distribution and density for the random variables
(a) Y = 1/(U + 1).
Attempt: I used a theorem F (Y)=P (Y<y) to get P ( 1/(U+1) is less than or equal to y), then
I got P ( -1-(1/y) is less than or equal to U is less than or equal to (1/y) -1), so Later I came
up with a CDF of F (y)=( 1/y) - 1 on [ 1/2,1 ] since it has to be uniform on [0,1]. However the cdf was 2-(1/y). How did they get that CDf?
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April 26th, 2017, 12:36 PM   #2
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$Y \sim \dfrac{1}{U+1}$

$\begin{align*}

&\phantom{=}F_Y(y)=P[Y<y] \\ \\

&=P[\dfrac{1}{U+1}<y] \\ \\

&=P[\dfrac 1 y - 1 < U] \\ \\

&=1 - Pr[U < \dfrac 1 y - 1] \\ \\

&=1 - \left(\dfrac 1 y - 1\right),~y \in \left[\dfrac 1 2,~1\right] \\

&=2-\dfrac 1 y,~y \in \left[\dfrac 1 2,~1\right]

\end{align*}$
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