Question

-wa exp{-(20 )2}, where The Normal(μ,02) distribution has density f(x) -oo < μ < oo and σ > 0. Let the randon variable T be such that X-log(T) is Normal(μ, σ2). Find the density of T. This distribution is known as the log normal Do not forget to indicate where the density of T is non-zero. 10.

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Answer #1

X = log T

of T = e^x

Theorem The exponentiation of a N(μ, σ2) random variable is a log normal(α, β) random variable. Proof Let the random variable X have the normal distribution with probability density function The transformation Y-g(X) = ex is a 1-1 transformation from χ = {x|-oo < x < oo} to y = {y| y > 0} with inverse X = g-1 (Y) = ln(Y) and Jacobian dX1 dY Y Therefore by the transformation technique, the probability density function of Y is In(u) )2 | 1 ys0 Let μ = ln(a) and σ = β. Then y > 0, which is the probability density function of the log normal distribution

pdf is non- zero for T > 0

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