Answer to Question #149682 in Statistics and Probability for Randal Rodriguez

Question

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 modeling these errors. Suppose the measurement error X of a certain physical quantity is decided by the density function

f(x)= {k(3-x^2), -1<x<1,
.........0,(included on the bracket) elsewhere

(a) Determine k that renders f(x) a valid density function. (Fraction form)
(b) Find the probability that a random error is more than 0. (Fraction form)
(c) Find the probability that a random error is within -1/2 and 1/2. (Fraction form)

Answer 1

A) $\int^1_{-1}f(x)dx=1$

$\int^1_{-1}k(3-x^2)dx=(3kx-kx^3/3)|_{-1}^1=3k-k/3+3k-k/3=6k-2k/3=18k/3-2k/3=16k/3=1$

k=3/16

B)$\int_0^1( 3(3-x^2)/16) dx=(9x/16-x^3/16)|_0^1=9/16-1/16=8/16=1/2$

C)$\int_{-1/2}^{1/2}(3 (3-x^2)/16) dx=(9x/16-x^3/16)|_{-1/2}^{1/2}=9/32-1/128+9/32-1/128=18/32-2/128=9/16-1/64=36/64-1/64=35/64$