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April 26th, 2017, 10:25 AM  #1 
Member Joined: Apr 2017 From: PA Posts: 35 Thanks: 0  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? 
April 26th, 2017, 12:36 PM  #2 
Senior Member Joined: Sep 2015 From: CA Posts: 1,265 Thanks: 650 
$Y \sim \dfrac{1}{U+1}$ $\begin{align*} &\phantom{=}F_Y(y)=P[Y<y] \\ \\ &=P[\dfrac{1}{U+1}<y] \\ \\ &=P[\dfrac 1 y  1 < U] \\ \\ &=1  Pr[U < \dfrac 1 y  1] \\ \\ &=1  \left(\dfrac 1 y  1\right),~y \in \left[\dfrac 1 2,~1\right] \\ &=2\dfrac 1 y,~y \in \left[\dfrac 1 2,~1\right] \end{align*}$ 

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