2.14.1 Lognormal

Question 1

What is a lognormal distribution?

Answer 1

A distribution where the logarithm of the variable is normally distributed.

Question 2

What’s the key relationship between the normal and lognormal distributions?

Answer 2

If X ~ Lognormal(μ, σ), then ln(X) ~ Normal(μ, σ)

Question 3

What is the PDF of a lognormal distribution?

Answer 3

f(x) = (1 / (xσ√(2π))) * exp( - (ln(x) - μ)² / (2σ²) ), for x > 0

Question 4

How is the CDF of a lognormal computed?

Answer 4

Use Φ((ln(x) - μ) / σ), where Φ is the standard normal CDF

Question 5

What is the mean of X ~ Lognormal(μ, σ)?

Answer 5

E[X] = exp(μ + σ² / 2)

Question 6

What is the variance of X ~ Lognormal(μ, σ)?

Answer 6

Var(X) = (exp(σ²) - 1) * exp(2μ + σ²)

Question 7

Are μ and σ the mean and standard deviation of X?

Answer 7

No — they are the mean and standard deviation of ln(X)

Question 8

What happens when you multiply independent lognormal variables?

Answer 8

Their product is also lognormal

Question 9

What parameters result from multiplying X_i ~ Lognormal(μ_i, σ_i)?

Answer 9

The product is Lognormal(Σμ_i, Σσ_i²)

Question 10

How do you calculate a percentile of a lognormal variable?

Answer 10

Take z such that Φ(z) = desired percentile → x = exp(μ + zσ)

Question 11

What trick helps identify a lognormal in compound expressions?

Answer 11

Look for exponentials of sums of normal variables

Question 12

How do you define a lognormal variable using a normal variable?

Answer 12

If Y ~ Normal(μ, σ), then X = e^Y is Lognormal(μ, σ)