Derive the probability density function of a lognormal random v... | Filo
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Consider the lognormal distribution with the density function given in Section 6.9. Suppose we have a random sample x1, x2, …, xn from a lognormal distribution. (a) Write out the likelihood function. (b) Develop the maximum likelihood estimators of μand σ^2. | Numerade
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SOLVED:Let X have a lognormal distribution with parameters (μ, σ^2). Show that E{X^r}=e^r μ (1)/(2) σ^2 r^2 and deduce that E{X}=e^μ (1)/(2) σ^2 and σX^2=e^2 μ σ^2(e^σ^2-1). (Hint: E{X^r}= ∫0^∞ x^r f(x) d x where f is the lognormal density; make the change of variables y=log(x)-μto obtain .E{X^r}=∫-∞^∞ (1)/(√(2 πσ^2)) e^(r μ r y-y^2 / 2 σ^2) d y .)
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