A thin tube of length one can be modeled by the interval 01

A thin tube of length one can be modeled by the interval (0,1). Suppose that the density of the material in the tube is given by ?(x)=32x^3-48x^2+22x. That is, if x denotes the distance from the left endpoint of the tube, the density at that point can be given by ?(x). We could, for convenience, define ?(x)=0 if x<0 or x>1.

A.) Observe that ? is not a probability density function. You can do this by calculating the mass M of the material occupying the thin tube.

B.) Rescale ? by a constant factor so that the result is a probability density function. Produce a graph of your result, let’s call it f=f(x).

C.) Now, using the PDF from part b), if X is RV that denotes the distance from the left endpoint of the tube, calculate E(X).

Solution