My Math Forum Compute $E(|X-Z|)$.

 July 4th, 2017, 04:21 PM #1 Member   Joined: Apr 2017 From: PA Posts: 45 Thanks: 0 Compute $E(|X-Z|)$. Let $X$ be a random variable distributed uniformly on [0,20]. Define a new random variable $Z$ by $Z$ = [X+.5 ] (the greatest integer in X). Find the expected value of $Z$. Compute $E(|X-Z|)$.

**Attempt:** I attempted to set up the integral $$\int_{0}^{20}\frac 1 {20}\,dx,$$ since I found out that it was uniformly distributed on $[0,1]$. Also, I thought the distribution fuction woud be $\dfrac 1 {20}$. Then, I used the formula $E(|X-Z|)$=E(X)-E(Z), I found out that $E(X)$=10, after doing the calculation, then I took the integral $$\int_{0}^{20}(X^2+.5X),dx,$$, but was unable to find the right answer.
 July 5th, 2017, 05:15 PM #2 Global Moderator   Joined: May 2007 Posts: 6,528 Thanks: 589 Z=0 for X < .5, |X-Z|=X Z=1 for .5
 July 6th, 2017, 10:01 AM #3 Member   Joined: Apr 2017 From: PA Posts: 45 Thanks: 0 So, we should stop at Z=10? Where 8.5
 July 6th, 2017, 01:17 PM #4 Global Moderator   Joined: May 2007 Posts: 6,528 Thanks: 589 Z=10 for (9.5
 July 6th, 2017, 02:38 PM #5 Member   Joined: Apr 2017 From: PA Posts: 45 Thanks: 0 So, we need to find out how much Z=? When 19.5
July 7th, 2017, 01:14 PM   #6
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Quote:
 Originally Posted by poopeyey2 So, we need to find out how much Z=? When 19.5
Z=20. Note that X stops at 20.

 July 10th, 2017, 12:12 PM #7 Member   Joined: Apr 2017 From: PA Posts: 45 Thanks: 0 So, we should stop at Z=19.5, where 19.5

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