Question

Find the density function of y = ez , where z ∼ n(µ, σ2). this is called the lognormal density, since log y is normally distributed.

Answer 1

Y = e^2 -> ln y = z [e^2 > 0 -> y > 0]

 

Fy (y) = P (Y ≤ y)

= P (e^2 £ y)

= P (Z ≤ ln y)

= Fx (ln y)

 

Differentiating, fy (y) = 1/y fz (ln y)

= 1/y * [(1 / σ sqrt of 2π) e^(1/2 ((ln y - µ)/a)^2)] , y > 0

 

Which is the required density function of y = e^2