A random variable Y has a lognormal distribution with parameters \mu and \sigma ^(2) if Y= exp(x) where x∼N(\mu ,\sigma ^(2)). Derive the probability density function of Y. Show your work.
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A random variable Y has a lognormal distribution with parameters $\mu$ and $\sigma^2$ if Y = exp(X) where X $\sim$ N($\mu$, $\sigma^2$). Derive the probability density function of Y. Show your work. Show more…
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If a random variable $U$ is normally distributed with mean $\mu$ and variance $\sigma^{2}$ and $Y=e^{U}$ [equivalently, $U=\ln (Y)]$, then $Y$ is said to have a log-normal distribution. The log-normal distribution is often used in the biological and physical sciences to model sizes, by volume or weight, of various quantities, such as crushed coal particles, bacteria colonies, and individual animals. Let $U$ and $Y$ be as stated. Show that a. the density function for $Y$ is $$f(y)=\left\{\begin{array}{ll} \left(\frac{1}{y \sigma \sqrt{2 \pi}}\right) e^{-(\ln y-\mu)^{2} /\left(2 \sigma^{2}\right)}, & y>0 \\ 0, & \text { elsewhere } \end{array}\right.$$ $$\text { b. } E(Y)=e^{\mu+\left(\sigma^{2} / 2\right)} \text { and } V(Y)=e^{2 \mu+\sigma^{2}\left(e^{x^{2}}-1\right)}$$
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