Circle pdf

What happens if this pdf is the arc of a circle?
Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving
Being curious Being resourceful Being resilient Being collaborative


A random variable $X$ has a zero probability of taking negative values but has a non-zero probability of taking values in the range $[0, a]$ for every $a>0$. The curve describing the probability density function forms an arc of a circle. Which of these are possible shapes (ignoring the scale) for the probability density function $f(x)$? Identify clearly the mathematical reasons, using the correct terminology, for your answers.

Image
Circle \pdf


If the radius of the circle forming the arc of the pdf is $1$, what is the maximum value that the random variable could possibly take?

Which of the other arcs are possible candidates for probability density functions? Can you invent mathematical scenarios which would lead to these pdfs?