Calculate the Cumulative density.

Question

$U$ is a random variable in the range of $(0,3)$. The random variable $W$ is the output of the clipper described by

$W=g(U)=U$ for $U\le 1$ and $1$ for $U>1$

find the cdf of $FW(w)$

Any ideas on how to solve this?

My idea was to graph it $g(U)$ vs $U$ and then see that the integral of the CDF is just area under the curve. However I think I am doing something wrong with my plot as I am getting the CDF$>1$?

"Cumulative density" is an oxymoron. The word "cumulative" contradicts the word "density". Probably you mean "cumulative distribution function". That's what "cdf" stands for. — Michael Hardy, May 13 '14 at 20:48
Is the distribution of U uniform (I would assume that from the symbol chosen, but it should be stated explicitly, if so)? — Rick Goldstein, May 13 '14 at 20:59

score 0 · Answer 1 · answered May 13 '14 at 21:45

If $U$ is a uniformly distributed random variable, then its density function equals $1/3$ on the given interval. Also, $W$ has a range from zero to 1 (even though its domain is the range of $U$. For all values of $w$ between 0 and 1, the probability density is $1/3$, and for 1, it's $2/3$.

So,

$$ F_W(w) = \begin{cases} 0, & w \le 0, \\ w/3, & 0 \lt w < 1, \\ 1, & w \ge 1 \end{cases} $$

Note the discontinuity.

Calculate the Cumulative density.

1 Answers1