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{\bf Question}

The cdf of a continuous r.v. $Y$ is given by

$$F(y)=\left\{\begin{array}{ll} 0, &amp; {\rm for}\
y&lt;0,\\2\frac{y^2}{\theta}, &amp; {\rm for}\ 0 \leq y \leq
\frac{\theta}{2},\\ 1-\frac{2(\theta-y)^2}{\theta^2}, &amp; {\rm for}\
\frac{\theta}{2} \leq y \leq \theta,\\ 1, &amp; {\rm for}\ y&gt;\theta,
\end{array} \right.$$

where $\theta$ is a positive constant.  Find the pdf of $Y$, and
sketch the cdf and the pdf of $Y$.

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{\bf Answer}

From the relationship between pdf an cdf

$$f(y)=\frac{dF(y)}{dy}=\left\{\begin{array}{ll} 0, &amp;
y&lt;0\\\frac{4y}{\theta^2}, &amp; 0\leq y\leq
\frac{\theta}{2}\\\frac{4(\theta-y)}{\theta^2}, &amp;
\frac{\theta}{2}\leq y\leq \theta\\0, &amp; \theta &lt;y \end{array}
\right.$$

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