Measurement error that is continuous and uniformly...

Name: Problem 60, Chapter 4: Applied Statistics and Probability for Engineers
Uploaded: 2021-08-19T03:00:10-08:00
Duration: 1 min 5 s
Channel: Yingtai Xiao
Description: Measurement error that is continuous and uniformly distributed from -3 to +3 millivolts is added to a circuit's true voltage. Then the measurement is rounded to the nearest millivolt so that it becomes discrete. Suppose that the true voltage is 250 millivolts. (a) What is the probability mass function of the measured voltage? (b) What are the mean and variance of the measured voltage?

Measurement error that is continuous and uniformly distributed from -3 to +3 millivolts is added to a circuit's true voltage. Then the measurement is rounded to the nearest millivolt so that it becomes discrete. Suppose that the true voltage is 250 millivolts. (a) What is the probability mass function of the measured voltage? (b) What are the mean and variance of the measured voltage?

Measurement error that is continuous and uniformly distributed from -3 to +3 millivolts is added to a circuit's true voltage. Then the measurement is rounded to the nearest millivolt so that it becomes discrete. Suppose that the true voltage is 250 millivolts.
(a) What is the probability mass function of the measured voltage?
(b) What are the mean and variance of the measured voltage?

Key Concepts

Uniform Distribution

A uniform distribution is a type of continuous probability distribution where all intervals of the same length within the support of the distribution are equally likely. In the context of measurement error, this means that the error, which is equally likely to take any value within the specified range, is modeled by a constant probability density function over that interval.

Rounding (Quantization)

Rounding, or quantization, is the process of converting a continuous random variable into a discrete random variable by mapping ranges of continuous values to specific discrete levels. This transformation alters the original distribution by essentially ‘binning’ continuous outcomes into discrete categories, which then have a corresponding probability mass function.

Probability Mass Function (PMF)

The probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. It is the discrete counterpart to the continuous probability density function, used after a transformation from a continuous to a discrete variable, such as when rounding a measurement error that was originally uniformly distributed.

Mean and Variance

The mean and variance are the first two moments of a probability distribution that respectively measure the central tendency and the spread of the distribution. When dealing with a discrete random variable obtained through rounding a continuous variable, the mean is the expected value computed as the sum of the possible outcomes weighted by their probabilities, while the variance quantifies the variability around this mean.

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