(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Consider a random variable, defined to bethe sum of two random real numbers chosen uniformly from [0, 1]. Let the random variables X and Y denote the two chosen real numbers. Define Z = X + Y . Derive the cumulative density function and the probability distribution function

2. The attempt at a solution

Ok really this is a problem in the text and I don't understand how they get certain things

Let me quote the parts that make and dont make sense to me

So far so good Here we take for our sample space the unit square in R2 with uniform density. A point [itex]\omega\epsilon\Omega[/itex] then consists of a pair (x, y) of numbers chosen at random. Then [itex]0\leq Z \leq 2[/itex]

The figure they refer to has a square whose sides has length 1. The region E.8 is a right angle triangle located at the bottom left of the square whose side lengths are 0.8. Why is it E.8? What does that mean?? Let [itex]E_{z}[/itex] denote the event that [itex]Z \leq z[/itex]. In Figure 2.14, we show the set E.8.

Ok why is that? First of all what is E.8 in the first place...? The event Ez, for any z between 0 and 1, looks very similar to the shaded set

in the figure.

For [itex]1 < z \leq 2 [/itex] the set Ez looks like the unit square with a triangle

removed from the upper right-hand corner.

We can now calculate the probability distribution [itex]F_{Z}[/itex] of Z; it is given by

[tex]F_{Z}(z)=P(Z\leq z)=\mbox{Area of}E_{z}[/tex]

[tex]f(x)=\left\{\begin{array}{cc}0,&\mbox{ if }

z< 0\\\frac{1}{2}z^2, & \mbox{ if } 0\leq z\leq 1\\ 1-\frac{1}{2}(2-z)^2 & \mbox{ if } 1\leq z\leq 2 \\ 1&\mbox{ if } z>2 \end{array}\right [/tex]

Why did they even break up the probability into 2 regions? Why can't 1 region suffice?

Also How did they come up with these equations? I understand how they came up with the regions which are 0 and 1. But as for the regions in between I am lost. Have they broken the unit square into 2 parts? The first part uses the region E.8? But why is the second region the way it is??

please Please help!! i am totally lost here...

Thank you in advance for your time and input

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# Homework Help: How to get Probability

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