The article "Modeling Sediment and Water Column Interactions...

The article "Modeling Sediment and Water Column Interactions for Hydrophobic Pollutants" (Water Res. $1984 : 1169-1174$ ) suggests the uniform distribution on the interval $[7.5,20]$ as a model for depth $(\mathrm{cm})$ of the bioturbation layer in sediment in a certain region. (a) What are the mean and variance of depth? (b) What is the cdf of depth? (c) What is the probability that observed depth is at most 10$?$ Between 10 and 15$?$ (d) What is the probability that the observed depth is within 1 standard deviation of the mean value?

The article "Modeling Sediment and Water Column Interactions for Hydrophobic Pollutants" (Water Res. $1984 : 1169-1174$ ) suggests the uniform distribution on the interval $[7.5,20]$ as a model for depth $(\mathrm{cm})$ of the bioturbation layer in sediment in a certain region.
(a) What are the mean and variance of depth?
(b) What is the cdf of depth?
(c) What is the probability that observed depth is at most 10$?$ Between 10 and 15$?$
(d) What is the probability that the observed depth is within 1 standard deviation of the mean value?

Key Concepts

Uniform Distribution

A uniform distribution is a type of continuous probability distribution where every outcome in a specified interval is equally likely to occur. This distribution is characterized by its constant probability density function over the interval, making it one of the simplest and most intuitive models for representing uncertainty when there is no reason to favor any particular outcome over another.

Mean and Variance of a Uniform Distribution

For a uniform distribution defined on the interval [a, b], the mean (or expected value) is calculated as (a + b)/2, which gives the central value of the distribution. The variance, which measures the spread of the data, is given by (b - a)²/12. These formulas provide fundamental statistical properties that help quantify the central tendency and variability of the uniform distribution.

Cumulative Distribution Function (CDF) of a Uniform Distribution

The cumulative distribution function (CDF) of a uniform distribution accumulates the probability from the lower bound up to any point within the interval. It increases linearly over the interval and is defined as 0 for values below a, (x - a)/(b - a) for values between a and b, and 1 for values above b. This linearity is a direct result of the uniform probability density across the interval.

Calculating Interval Probabilities

In a uniform distribution, the probability that a random variable falls within any sub-interval is simply the length of that sub-interval divided by the total length of the interval [a, b]. This property makes computing probabilities over specific ranges straightforward because it relies solely on the proportion of the sub-interval’s length to the overall interval.

Standard Deviation and Its Use in Probability

The standard deviation is the square root of the variance and provides a measure of dispersion or spread around the mean. In the context of continuous distributions like the uniform distribution, determining the probability that a variable falls within one standard deviation of the mean involves identifying the interval corresponding to this measure and then calculating the proportion of the total interval it represents. This concept is useful for understanding the concentration of probability mass around the central value.

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