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To understand the mean and variance of a uniform distribution, it's essential to grasp that this type of distribution is about equal probabilities. In our exercise, the uniform distribution is between 7.5 and 20 cm for a specific environmental observation.

The mean is the average value you would get if you collected a large number of data points from this distribution. With a formula, calculating the mean \( \mu \) of a uniform distribution across interval \((a, b)\) is straightforward:

• Mean \( \mu = \frac{a + b}{2} \)

For our example, substituting the given values:

• \( \mu = \frac{7.5 + 20}{2} = 13.75\)

Hence, the mean depth is 13.75 cm.

Variance measures how spread out the values of the distribution are. For a uniform distribution, you can compute the variance \( \sigma^2 \) with this formula:

• Variance \( \sigma^2 = \frac{(b-a)^2}{12} \)

Plugging numbers into the equation for our interval:

• \( \sigma^2 = \frac{(20 - 7.5)^2}{12} = 13.0208333\)

So, the variance of the depth is approximately 13.02. This means there is a moderate spread of data around the mean.

The Cumulative Distribution Function (CDF) is a key tool in understanding probabilities within a specific range of a uniform distribution. It helps determine the probability that a random variable is less than or equal to a certain value.

For a uniform distribution over the interval \((a, b)\), the CDF \( F(x) \) is defined as follows:

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

In the context of the exercise:

• For \( x < 7.5 \), the probability is 0 because it's outside the range.
• For \( 7.5 \leq x \leq 20 \), \( F(x) = \frac{x - 7.5}{12.5} \)
• For \( x > 20 \), the entire probability is captured, so \( F(x) = 1 \)

CDF provides a complete view of how values accumulate across the distribution, reflecting how likely a depth is to fall within certain values.

The standard deviation in any distribution tells us about the average distance of data points from the mean. In a uniform distribution, it remains a useful measure to understand the spread.

The standard deviation \( \sigma \) is the square root of the variance. Calculating this gives us:

• \( \sigma = \sqrt{13.0208333} \approx 3.61 \)

With this, we can find out how many depths are likely within certain distances from the mean. For one standard deviation (\( \, \mu - \sigma \, \) to \( \, \mu + \sigma \,\)), calculate:

• \( 13.75 - 3.61 = 10.14 \)
• \( 13.75 + 3.61 = 17.36 \)

This range covers a probability of about 0.576 or 57.6%.

For two standard deviations, going further:

• \( 13.75 - 2 \times 3.61 = 6.53 \)
• \( 13.75 + 2 \times 3.61 = 20.97 \)

Since these bounds spill beyond the interval limits (7.5 to 20), this range covers the entire valid range of data, giving a probability of 1 or 100%.

Knowing these probabilities helps infer how typical or unusual certain depth measurements might be when compared to the expected average.