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Wikipedia has a nice article on log-normal distributions: https://en.m.wikipedia.org/wiki/Log-normal_distribution. The article reveals that the log-normally distributed X and the normally distributed log(X) have different means and standard deviations. If X follows a log-normal distribution with parameters $\mu$ and $\sigma$, then $\mu$ and $\sigma$ ...


6

assume X is lognormally distributed and Y is normally distributed where Y = log(X) This is where you are confused. You don't make assumptions on two distributions, one of which just happens to be the log of the other. Instead, you start with a distribution $X$. Then you consider $\log X$. If $\log X\sim N(\mu,\sigma^2)$, then we say that the original ...


4

The Shapiro-Wilk test considers how closely a probability plot of the data adheres to an ideal diagonal line. For instance, here are probability plots of four independent samples of size $n=5$ from a Normal$(10,2^2)$ distribution: Recall that a probability plot is a scatterplot of the points $(m_i, x_{(i)})$ where $x_{(1)}\le x_{(2)}\le \cdots \le x_{(n)}$ ...


1

Specific case: Let $X \sim \mathsf{Norm}(\mu_X = 69, \sigma_X = 4)$ and, independently, $Y \sim \mathsf{Norm}(\mu_Y = 66, \sigma_Y = 3).$ I'm not saying this is exactly correct for any populations of men and women (in inches), but it can serve as an example. Then, following @MartijnWettering's suggestion, let $D = X - Y,$ so that $\mu_D = 3$ and $\sigma_D ...


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In addition to the excellent answers already posted, I thought it might be helpful to have a visualization exploring the distributions of observed proportions for varying $n$ and $p$ values. To generate the below histograms, I took $n$ samples from a Bernoulli trial with probability $p$, and repeated this process 10,000 times. I then generated a histogram ...


1

The answer by Glen_b frames the HPD as a set of simultaneous equations that can be solved via numerical methods. This is one possible way to compute the HPD. An alternative method is to frame the HPD as an optimisation problem, and solve this via numerical methods. Computing the HDR via optimisation: Suppose you have a random variable $X \sim f$ where the ...


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