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In statistics, the mean is the average value of a data set, providing a central point around which other data points distribute. For a uniform distribution on an interval \(a, b\), calculating the mean involves a simple formula: \( \mu = \frac{a + b}{2} \). This formula finds the midpoint of the interval, representing the "center" of the distribution.
In this exercise, with the interval \(7.5, 20\), the mean becomes \( \mu = \frac{7.5 + 20}{2} = 13.75 \).

On the other hand, the variance measures how much the data varies around the mean. For a uniform distribution, the variance \(\sigma^2\) is given by: \(\sigma^2 = \frac{(b-a)^2}{12}\). It reflects the spread of the data points; a higher variance means data points are more spread out from the mean.
Here, the variance is computed as: \( \sigma^2 = \frac{(20 - 7.5)^2}{12} = 13.02\). This indicates how the depths within the interval are distributed around the average depth of 13.75 cm.

The Cumulative Distribution Function (CDF) helps us understand how probabilities accumulate over a range of values in a distribution. For a uniform distribution \(U(a, b)\), the CDF is expressed mathematically as a piecewise function:

• \(F(x) = 0, \text{ for } x < a\)
• \(F(x) = \frac{x-a}{b-a}, \text{ for } a \leq x \leq b\)
• \(F(x) = 1, \text{ for } x > b\)

This function helps in understanding the probability that the variable, say depth \(X\), will take a value less than or equal to \(x\).
Applying this to our interval \(\[7.5, 20\]\), the CDF becomes:

• \(F(x) = 0, \text{ when } x < 7.5\)
• \(F(x) = \frac{x-7.5}{12.5}, \text{ when } 7.5 \leq x \leq 20\)
• \(F(x) = 1, \text{ when } x > 20\)

The CDF provides a simple yet powerful tool for determining probabilities within specified intervals.

Probability calculations in statistics help us quantify the likelihood of a specific outcome in an experiment. Utilizing the cumulative distribution function (CDF), we can determine probabilities for a range of values within a uniform distribution.
For example, to calculate the probability of the depth being at most 10cm, evaluate the CDF at 10: \(P(X \leq 10) = F(10) = \frac{10-7.5}{12.5} = 0.2\).

To find the probability of depth between 10 and 15cm, we need the difference in CDF values: \(P(10 < X \leq 15) = F(15) - F(10) = 0.4\). This is helpful for gauging likelihoods within specific ranges.

Lastly, evaluating probabilities within standard deviations involves using both mean and variance. For one standard deviation, the depth range covers \(10.14\) to \(17.36\) cm. The probability is \(P(10.14 < X < 17.36) = 0.5736\). Similarly, the two standard deviation range covers from \(7.5\) to \(20\) cm, resulting in a probability of \(1\) or certainty that a depth falls within this range. These calculations give insights into how often certain conditions are met.