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## Q. 2.3

Let the random variable X represent the DC voltage output from some force transducer, and let the distribution be uniform on the set [-4,16] such that $F_{X}(u)=0.05 (u+4)$ for $-4 \leq u \leq 16$. Of course, $F_{X}(u)$  is zero to the left and unity to the right of the [-4,16] interval. Now let another (mixed) random variable Y represent the output from a voltmeter that reads only from zero to 10 volts and that has X as the input. Whenever $0 \leq X \leq 10$ we will have Y = X, but we will get Y=0 whenever $X \lt 0$, and Y=10 whenever $X \gt 10$. Find the cumulative distribution function for Y.

## Verified Solution

From $P(Y \leq u)$ we get

 $F_{Y}(u)=0$ For $u \lt 0$ $F_{Y}(u)=0.05(u+4)$ For $0 \leq u \lt 10$ $F_{X}(u)=1$ For $u \geq 10$

This $F_{Y}(u)$ function for this mixed random variable has discontinuities of 0.2 at $u=0$ and 0.3 at $u=10$, representing the finite probabilities that Y takes on these two particular values.