Question $1$

Let $x$ be a random variable with density

$f(x)=\{\begin{array}{cc}C(4x-x^2),& 0x3\\ 0,& \text{o}\text{t}\text{h}\text{e}\text{r}\text{w}\text{i}\text{s}\text{e}& \end{array}$

where CinR is some constant.

Draw the graph of this piece$-$wise density function. This always helps a great deal, when computing values of various atributes of the distribution

of $x.$ It also helps to spot a possible error in computation, since often you can tell based on the graph of the density function whether certain

computed value is plausible or doesn't make any sense.

Part a$)$

Find the constant $C.$

Write your answer as

$C=$

For example, if you think the answer is $\frac{12}{34}$ you should create variable $C$ as

IMPORTANT: Just typing $\frac{12}{34},$ or answer $=\frac{12}{34}$ is not acceptable. Your answer will be graded automatically, and the test code expects you to

create a Python variable called $C,$ and the code will test the value of $C.$ Do not round your answer.Part b$)$

Find the cumulative distribution function $($cdf$)$ of $x.$ Create a python function $F(x),$ which for any input xinR, returns the value of the cdf$,$ i$.$e$.P(xx).$

Do NOT round the output value.