Measurements of scientific systems are always subject to variation, some more than others. There are many structures for measurement error, and statisticians spend a great deal of time modelling these errors. Suppose the measurement error X of a certain physical quantity is decided by the density function ?(?) = {?(3 − ?2) −1 < ? < 1 0 ?????ℎ??? i) Determine k that renders f(x) a valid density function. ii) Find the probability that a random error in measurement is more than 0.3.

Help with parts c thru f

Prove that d-function is the distribution derivative of the Heaviside function { if x > 0, if x < 0. H(x) =

3.30 Measurements of scientific systems are always subject to variation, some more than others. There are many structures for measurement error, and statis- ticians spend a great deal of time modeling these errors. Suppose the measurement error X of a certain physical quantity is decided by the density function [k(3-x²), f(x) = 10, (a) Determine k that renders f(x) a valid density func- tion. (b) Find the probability that a random error in mea- surement is less than 1/2. −1≤x≤1, elsewhere. (c) For this particular measurement, it is undesirable if the magnitude of the error (i.e., |x|) exceeds 0.8. What is the probability that this occurs?