Suppose a cw r$.$v$.$ is distributed triangularly over the range $(-$M$,$ M$)$ as the density function

p$($x$)=$ P$0(1-|$x $|).$ Note that probability is always nonnegative.

$($i$)$ Find P$0.$ Show your work.

$($ii$)$ According to the result in $($i$),$ is there a physical limit on M$?$ Explain.

$($iii$)$ If M is allowed the range according to $($ii$),$ would p$($x$)$ have physical meaning? Hence, what should an acceptable maximum value of M$?$ Explain.

$($iv$)$ Write down an expression for the differential entropy of this r$.$v$.,$ and simplify the integral as far as possible. Do not solve the integral.

Guidelines: Note that the density function cannot be negative, and the area under it must be $1.$ This will determine the limits of M$.$