Question

Suppose that the measurement error X of a certain physical quantity is computed by the following density function. f(x)={ k(4−x 3 ),0,−1≤x≤1 elsewhere

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a) Determine k that renders f(x) a valid probability distribution 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 exceeds 0.8 . What is the probability that this occurs?

Answer 1

To determine the normalization constant k, integrate the probability density function f(x) over the interval -1 to 1 and set it equal to 1. Solving for k gives the value 4/15, which renders f(x) a valid probability distribution function.

Step-by-step explanation:

The probability density function f(x) for the measurement error X is given by:

f(x) = k(4 - x^3), -1 ≤ x ≤ 1

We need to determine the value of k that makes f(x) a valid probability distribution function. To do this, we need to normalize f(x) such that the total area under the curve is equal to 1.

To find the normalization constant k, we integrate f(x) over the interval -1 to 1 and set it equal to 1:

∫[-1]1k(4 - x^3)dx = 1

Integrating this expression gives:

∫[-1]1k(4 - x^3)dx = k[4x - (x^4/4)] |[-1]1 = 1

Substituting the limits of integration and solving for k gives:

k(4 - 1/4) - k(-4 + 1/4) = 1

Simplifying this equation, we get:

15k/4 = 1

k = 4/15

Therefore, the value of k that renders f(x) a valid probability distribution function is 4/15.