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I work through a simple Python bvimplementation of geometric Brownian motion and check it against the theoretical model.

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In probability theory, Bienaymé's identity is a formula for the variance of random variables which are themselves sums of random variables. I provide a little intuition for the identity and then prove it.

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I derive some basic properties of the lognormal distribution.

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A useful view of a covariance matrix is that it is a natural generalization of variance to higher dimensions. I explore this idea.

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I discuss moving or rolling averages, which are algorithms to compute means over different subsets of sequential data.

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I discuss and prove the Gauss–Markov theorem, which states that under certain conditions, the least squares estimator is the minimum-variance linear unbiased estimator of the model parameters.

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How do we know when a parameter estimate from a random sample is significant? I discuss the use of standard errors and confidence intervals to answer this question.

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I provide a numerical demonstration that the mutual information of two random variables, the observations and latent variables in a Gaussian mixture model, is symmetric.

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The mutual information (MI) of two random variables quantifies how much information (in bits or nats) is obtained about one random variable by observing the other. I discuss MI and show it is symmetric.

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I derive the entropy for the univariate and multivariate Gaussian distributions.

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Why are a distribution's moments called "moments"? How does the equation for a moment capture the shape of a distribution? Why do we typically only study four moments? I explore these and other questions in detail.

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Under certain regularity conditions, maximum likelihood estimators are "asymptotically efficient", meaning that they achieve the Cramér–Rao lower bound in the limit. I discuss this result.

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The Cramér–Rao lower bound allows us to derive uniformly minimum–variance unbiased estimators by finding unbiased estimators that achieve this bound. I derive the main result.

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I document several properties of the Fisher information or the variance of the derivative of the log likelihood.

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I walk the reader through a proof the Rao–Blackwell Theorem.

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I discuss a straightforward proof of the law of total expectation with three standard assumptions.

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I show how several properties of the distribution of a random variable—the expectation, conditional expectation, and median—can be viewed as solutions to optimization problems.

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Probability distributions that are members of the exponential family have mathematically convenient properties for Bayesian inference. I provide the general form, work through several examples, and discuss several important properties.

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Many probabilistic models assume random noise is Gaussian distributed. I explain at least part of the motivation for this, which is grounded in the Central Limit Theorem.

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The KL divergence, also known as "relative entropy", is a commonly used metric for density estimation. I re-derive the relationships between probabilities, entropy, and relative entropy for quantifying similarity between distributions.

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Bessel's correction is the division of the sample variance by $N - 1$ rather than $N$. I walk the reader through a quick proof that this correction results in an unbiased estimator of the population variance.

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