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July 4th, 2017, 04:21 PM   #1
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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 $E(|X-Z|)$.

<br><br>
**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.
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July 5th, 2017, 05:15 PM   #2
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Z=0 for X < .5, |X-Z|=X
Z=1 for .5<X<1.5, |X-Z| is uniform in the interval (0,.5)
Z=2, etc. leads to |X-Z| uniform in the interval (0,.5)

You should be able to do the rest.
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July 6th, 2017, 10:01 AM   #3
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So, we should stop at Z=10? Where 8.5<X<9.5?
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July 6th, 2017, 01:17 PM   #4
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Z=10 for (9.5<X<10.5). Your original statement has X up to 20.
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July 6th, 2017, 02:38 PM   #5
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So, we need to find out how much Z=? When 19.5<X <20.5
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July 7th, 2017, 01:14 PM   #6
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Quote:
Originally Posted by poopeyey2 View Post
So, we need to find out how much Z=? When 19.5<X <20.5
Z=20. Note that X stops at 20.
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July 10th, 2017, 12:12 PM   #7
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So, we should stop at Z=19.5, where 19.5<X <20?
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