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:

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.

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)

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.

(a) The value of k that makes f(x) a valid density function is k = 1/6.

(b) The probability that a random error in measurement is less than 1/2 is 3/4.

(c) The probability that the magnitude of the error exceeds 0.8 is 1/4.

(a) To make the given function f(x) a valid probability density function, it must satisfy the following conditions:

The function must be non-negative for all x: f(x) ≥ 0.

The total area under the probability density function must equal 1: ∫f(x)dx from -1 to 1 = 1.

Given , -1 ≤ x ≤ 1, and f(x) = 0 elsewhere, let's find the value of k that satisfies these conditions.

Non-negativity: The function is non-negative for -1 ≤ x ≤ 1, so we have ≥ 0 for -1 ≤ x ≤ 1. This means that k can be any positive constant.

Total area under the probability density function: To find the value of k, integrate f(x) over the interval [-1, 1] and set it equal to 1:

∫[from -1 to 1] = 1

∫[-1, 1] = 1

Now, integrate the function:

from -1 to 1 = 1

Simplify:

[3k - k/3 + 3k + k/3] = 1

6k = 1

k = 1/6

So, the value of k that makes f(x) a valid density function is k = 1/6.

(b) To find the probability that a random error in measurement is less than 1/2, you need to calculate the integral of f(x) from -1/2 to 1/2:

P(-1/2 ≤ X ≤ 1/2) = ∫[from -1/2 to 1/2] f(x)dx

P(-1/2 ≤ X ≤ 1/2) = ∫[-1/2, 1/2] (1/6)

Now, integrate the function:

from -1/2 to 1/2

Simplify:

(1/6)[(3/2 - 1/24) - (-3/2 + 1/24)]

(1/6)[(9/8) + (9/8)]

(1/6)(18/8)

(3/4)

So, the probability that a randomerror in measurement is less than 1/2 is 3/4.

(c) To find the probability that the magnitude of theerror (|x|) exceeds 0.8, you need to calculate the probability that |X| > 0.8. This is the complement of the probability that |X| ≤ 0.8, which you can calculate as:

P(|X| > 0.8) = 1 - P(|X| ≤ 0.8)

P(|X| > 0.8) = 1 - P(-0.8 ≤ X ≤ 0.8)

We already found P(-0.8 ≤ X ≤ 0.8) in part (b) to be 3/4, so:

P(|X| > 0.8) = 1 - 3/4

P(|X| > 0.8) = 1/4

So, the probability that the magnitude of the error exceeds 0.8 is 1/4.

To Learn more about probability here:

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