Question

3.30 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 − x2), −1 ≤ x ≤ 1, 0, elsewhere. (a) Determine k that renders f(x) a valid density function. (b) Find the probability that a random error in measurement is less than 1/2. (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?

Answer 1

Answer:

a) k should be equal to 3/16 in order for f to be a density function.

b) The probability that the measurement of a random error is less than 1/2 is 0.7734

c) The probability that the magnitude of a random error is more than 0.8 is 0.164

Step-by-step explanation:

a) In order to find k we need to integrate f between -1 and 1 and equalize the result to 1, so that f is a density function.

$1 = k \int\limits^1_{-1} {(3-x^2)} \, dx = k * (3x-\frac{x^3}{3})|_{x=-1}^{x = 1} = k*[(3-1/3) - (-3 + 1/3)] = 16k/3$

16k/3 = 1

k = 3/16

b) For this probability we have to integrate f between -1 and 0.5 (since f takes the value 0 for lower values than -1)

$P(X < 1/2) = \int\limits^{0.5}_{-1} {\frac{3}{16}(3-x^2)} \, dx = \frac{3}{16} [(3x-\frac{x^3}{3}) |_{x=-1}^{x=0.5}] =\frac{3}{16} *(1.458333 - (-3+1/3)) = 0.7734$

c) For |x| to be greater than 0.8, either x>0.8 or x < -0.8. We should integrate f between 0.8 and 1, because we want values greater than 0.8, and f is 0 after 1; and between -1 and 0.8.

$P(|X| > 0.8) = \int\limits^{-0.8}_{-1} {\frac{3}{16}*(3-x^2)} \, dx + \int\limits^{1}_{0.8} {\frac{3}{16}*(3-x^2)} \, dx =\\ \frac{3}{16} (3x-\frac{x^3}{3})|_{x=-1}^{x=-0.8} + \frac{3}{16} (3x-\frac{x^3}{3})|_{x=0.8}^{x=1} = 0.082 + 0.082 = 0.164$