Final answer:
To find the mean and standard deviation for SO₂ concentration with a lognormal distribution, you calculate the mean using the formula exp(μ + (σ²/2)) and the standard deviation using the formula [(exp(σ²) - 1) * exp(2μ + σ²)]¹⁰⁵, substituting the given values of μ = 1.9 and σ = 0.7.
Step-by-step explanation:
Understanding Lognormal Distribution
The lognormal distribution is used to describe the distribution of a variable whose logarithm is normally distributed. Given the parameters μ = 1.9 and σ = 0.7 for the lognormal distribution of SO₂ concentrations above a forest, we can find the mean and standard deviation of the concentration.
Calculating the Mean
The mean of a lognormal distribution is given by the formula: E(X) = exp(μ + (σ²/2)). Substituting μ = 1.9 and σ = 0.7 into the formula, we get:
E(X) = exp(1.9 + (0.7² / 2))
= exp(1.9 + 0.245)
= exp(2.145)
The approximate value is then calculated using a calculator.
Calculating the Standard Deviation
The standard deviation of a lognormal distribution is given by: σ(X) = [(exp(σ²) - 1) * exp(2μ + σ²)]¹⁰⁵. Again using μ = 1.9 and σ = 0.7:
σ(X) = [(exp(0.7²) - 1) * exp(2 * 1.9 + 0.7²)]¹⁰⁵
= [(exp(0.49) - 1) * exp(3.8 + 0.49)]¹⁰⁵
The approximate value is obtained with a calculator.
The mean value and standard deviation are important statistical measures that describe the center and spread of a distribution, respectively.