Assume that the wavelengths of photosynthetically active radiations (PAR) are uniformly distributed at integer nanometers in the red spectrum rom 675 to 700 nm. 1. What is the mean of the wavelength distribution for this radiation? 2. What is the variance of the wavelength distribution for this radiation? 3. If instead, the wavelengths are uniformly distributed at integer nanometers from 75 to 100 nanometers, how do the mean and variance of the wavelength distribution compare to the original distribution? Briefly explain.

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Explanation:

1. What is the mean of the wavelength distribution for this radiation?

Mean = (a+b)/2

= (675 + 700)/2

= 1375/2

= 687.5

2. What is the variance of the wavelength distribution for this radiation?

Variance= {(b-a+1)^2 - 1}/12

= {(700 - 675 + 1)-1}^2/12

= (26^2)-1/12

= 676-1/12

= 675/12

= 56.25

3. If instead, the wavelengths are uniformly distributed at integer nanometers from 75 to 100 nanometers, how do the mean and variance of the wavelength distribution compare to the original distribution?

New Mean = (a+b)/2

= (75 + 200)/2

= 175/2

= 87.5

New Variance= {(b-a+1)^2 - 1}/12

= {(100 - 75 + 1)-1}^2/12

= (26^2)-1/12

= 676-1/12

= 675/12

= 56.25

From the solutions, while the mean differs, the variance (56.25) is thesame due to the fact that variance depends on interval length and the first and second question has same interval length.

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