Question

A) Determine the probability that the measurement error X takes a value less than 2/3 using the given density function:

Calculate ∫(from -1 to 2/3) f(x) dx where f(x) is defined as f(x) = (28/3)(5 - x^2) for -1 ≤ x ≤ 1, and 0 otherwise.

b) Find the mean and standard deviation of X based on the provided density function. Please provide your calculations or methods for both the mean and standard deviation.

c) Determine the distribution function F(x) for the measurement error X. Please outline the steps to find this distribution function.

Answer 1

To answer the question, integrate the density function to find the probability, use statistical formulas to calculate the mean and standard deviation, and integrate the density function to define the cumulative distribution function.

Step-by-step explanation:

To solve part a, we integrate the density function from -1 to 2/3, which requires the use of calculus and the specific density function provided. For parts b and c, we employ statistical formulas to calculate the mean and standard deviation and to define the cumulative distribution function (CDF), respectively.

To answer the student in terms of their specifics:

1. To find the probability distribution of X: We calculate the integral of the density function over the specified range to obtain the probability that X is less than 2/3.
2. To calculate the mean and standard deviation: We use the formulas for the expectation and variance of a continuous random variable, where the mean (expected value) is the integral over all possible values of x times the density function, and the standard deviation is the square root of the variance.
3. To determine the distribution function F(x): We integrate the density function from the lower bound of x (in this case, -1) to any given value of x to find the CDF F(x).