Wavelength variance

Question

Assume that the wavelengths of photosynthetically active radiations (PAR) are uniformly distributed at integer nanometers in the red spectrum from 640 to 670nm.

What is the variance of the wavelength distrbution for this radiation?

Answer 1

Hi Al,

I just joined the platform, which is why my response is a bit late :)

Anyways, if my answer below is helpful, I would greatly appreciate your feedback!

Final answer: 2000 nanometers squared (nm²)

To calculate the variance of the wavelength distribution for photosynthetically active radiations (PAR) in the red spectrum, which is uniformly distributed at integer nanometers from 640 to 670nm, you can use the formula for the variance of a discrete uniform distribution.

In a discrete uniform distribution, each possible outcome (in this case, each integer nanometer wavelength) has equal probability.

Here's the formula for the variance of a discrete uniform distribution:

Variance (σ²) = [(n² - 1) * (b - a)] / 12

Where:

1. n is the number of possible outcomes (integer nanometers from 640 to 670, so n = 670 - 640 + 1 = 31)
2. a is the minimum value (640 nanometers)
3. b is the maximum value (670 nanometers)

Now, plug in these values:

σ² = [(31² - 1) * (670 - 640)] / 12

σ² = [(961 - 1) * 30] / 12

σ² = [960 * 30] / 12

σ² = 24000 / 12

σ² = 2000

So, the variance of the wavelength distribution for photosynthetically active radiations (PAR) in the red spectrum from 640 to 670nm is 2000 nanometers squared (nm²).

Hope this helps!

Thank you,

Benjamin M.