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The mean of a uniform distribution is quite straightforward to calculate. In a uniform distribution, probabilities are spread evenly over a certain range of values. The formula to find the mean of a uniform distribution is \( \mu = \frac{a + b}{2} \),where \( a \) and \( b \) are the minimum and maximum values of the distribution, respectively. The mean, also known as the average, tells us the central value of this distribution.

• In part (a) of our exercise, the wavelengths for photosynthetically active radiations ranged from 675 nm to 700 nm.Using the formula here, the mean is calculated as \( \mu = \frac{675 + 700}{2} = 687.5 \ \text{nm}\).
• In part (b), the range was from 75 nm to 100 nm, resulting in a mean of \( \mu = \frac{75 + 100}{2} = 87.5 \ \text{nm}\).

This difference shows that while the spread (or variance) may be the same, the average or center point of the wavelengths can shift significantly with different ranges.

Variance in a uniform distribution tells us about the spread or how much the values deviate from the mean. The formula for variance in a uniform distribution is \( \sigma^2 = \frac{(b - a + 1)^2 - 1}{12} \).The variance shows us how dispersed the values are across the range — a higher variance indicates a wider spread.

• For the wavelengths distribution in part (a) between 675 nm and 700 nm, the variance was calculated as \( \sigma^2 = \frac{(700 - 675 + 1)^2 - 1}{12} = 56.33 \).
• For part (b), from 75 nm to 100 nm, the variance calculated the same because the numbers of integer values and the difference between \( b \) and \( a \) remained constant.This reinforces that variance is dependent not just on the absolute values but primarily on the range size and count of values in a uniform distribution.

The key takeaway is that variance measures how much the values vary from the mean within the specified range.

Understanding that wavelengths can be uniformly distributed is crucial in studies of light and radiation such as photosynthetically active radiations (PAR). Uniform distribution implies that each wavelength within a specified range has an equal probability of occurring. In practical terms for this exercise, whether we consider red spectrum wavelengths from 675 to 700 nm or a range from 75 to 100 nm, the characteristic of uniform distribution remains consistent. It means each individual wavelength has the same chance of being observed within its respective range.

• This exercise demonstrated that regardless of the wavelength range, if it’s uniformly distributed, the way we calculate mean and variance doesn’t change.
• Although the specific wavelengths differ, the methods and mathematical formulas applied are universals for any uniformly distributed data set.

Uniform distribution helps simplify complex realities of natural phenomena, providing a base to apply statistical methods in diverse fields such as physics, chemistry, and environmental science.