Let $X$ and $Y$ be two independent random variables having the same uniform distribution $U(0,1)$ with density

$f(x)=1$ if $0≤x≤1$ (and $0$ elsewhere).

Let $Z$ be a real random variable defined by:

$Z=X-Y$ if $X>Y$ (and $0$ elsewhere).

  1. Derive the distribution of $Z$.

  2. Compute the expectation $E(Z)$ and variance $V(Z)$.

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    $\begingroup$ Homework? What have you tried and where are you stuck? Do you know how to find the distribution of a sum of independent random variables. If you do then, hint: $X - Y = X + (-Y)$. That said, your question doesn't seem to be asking about the distribution of a (pure) subtraction. So, providing some details on your thought process will help users here guide you in the right direction. $\endgroup$ – cardinal Nov 9 '11 at 17:44
  • $\begingroup$ i am preparing for an exam after leaving university for 5 years and working in a totally different field that has nothing to do even with numbers. $\endgroup$ – Majed Hijazi Nov 9 '11 at 18:15
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    $\begingroup$ The distribution is more than just the mean and variance, so you should review the distinction among the three. Then consider relying on first principles. E.g., drawing a picture of the joint distribution of $(X,Y)$ in the $x,y$-plane along with level curves of $Z=X-Y$ will provide an immediate (and easy) geometric derivation of the distribution of $Z$. $\endgroup$ – whuber Nov 9 '11 at 18:29
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    $\begingroup$ Is this exactly how the problem is originally stated? Or does the problem ask for the conditional distribution of $Z$ given that $X\gt Y$? $\endgroup$ – whuber Nov 9 '11 at 18:34
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    $\begingroup$ Hint: Since $P\{X < Y\} = \frac{1}{2}$ (think about why this must be so), $Z$ has value $0$ with probability $\frac{1}{2}$. Thus, $Z$ is what is sometimes called a mixed random variable which takes on some values with nonzero probability and behaves as a continuous random variable for some values. As @whuber does, I too ask whether you have mis-stated the problem. It leads to more complications than one would expect from a typical end-of-chapter problem at the apparent level of the book you are using. $\endgroup$ – Dilip Sarwate Nov 10 '11 at 12:50

Check p. 18 of Probability Distributions as Program Variables, by Dimitrios Milios.

He has discussed the problem in an fairly detailed manner.


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