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

Question

Mathematics

Ashleigh Mcmahon

6 December, 13:38

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

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Tatiana · 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