Please show steps. Thank You! Measurements of scientific systems are always subj
ID: 3131949 • Letter: P
Question
Please show steps. Thank You!
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 Inequality x Inequality 1 Determine the k that renders f(x) a valid density function 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? Use the cdf to calculate the probability.Explanation / Answer
A)
f(x) = k(3-x^2)
For valid density function :
integral from (-1 -> 1) k(3-x^2)dx = 1
=> from (-1 -> 1) k(3x - x^3/3) = 1
=> k((3 -1/3)- (-3+1/3)) =1
=> k(8/3 + 8/3) = 1
=> k = 3/16 Answer
B)
P(X<=0.8) = integral from (-1 -> 0.8) 3/16 * (3-x^2)dx
=> 3/16 * from (-1 -> 0.8) (3x - x^3/3)
=> 3/16 * ((3*0.8 -(0.8)^3/3)- (-3+1/3))
=> 0.918 Answer