Posts Tagged ‘ mathematics ’

Mathematics and Mathematical Statistics Lesson of the Day – Convex Functions and Jensen’s Inequality

Mathematics and Mathematical Statistics Lesson of the Day – Convex Functions and Jensen’s Inequality

Consider a real-valued function that is continuous on the interval , where and are any 2 points in the domain of .  Let be the midpoint of and .  Then, if then is defined to be midpoint convex. More generally, let’s consider any point within the interval .  We can denote this arbitrary point as where . […]

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Mathematical Statistics Lesson of the Day – The Glivenko-Cantelli Theorem

Mathematical Statistics Lesson of the Day – The Glivenko-Cantelli Theorem

In 2 earlier tutorials that focused on exploratory data analysis in statistics, I introduced the conceptual background behind empirical cumulative distribution functions (empirical CDFs) how to plot  empirical cumulative distribution functions in 2 different ways in R There is actually an elegant theorem that provides a rigorous basis for using empirical CDFs to estimate the true CDF – and […]

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Lebesgue Measure and Outer Measure Problems

September 8, 2014
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More proving, still on Real Analysis. This is my solution and if you find any errors, do let me know.ProblemsLebesgue Measure: Let $\mu$ be set function defined for all set in $\sigma$-algebra $\mathscr{F}$ with values in $[0,\infty]$. Assume $\mu$ is ...

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Translation Invariant of Lebesgue Outer Measure

September 7, 2014
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Another proving problem, this time on Real Analysis.ProblemProve that the Lebesgue outer measure is translation invariant. (Use the property that, the length of an interval $l$ is translation invariant.) SolutionProof. The outer measure is translation ...

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Mathematical and Applied Statistics Lesson of the Day – The Motivation and Intuition Behind Chebyshev’s Inequality

Mathematical and Applied Statistics Lesson of the Day – The Motivation and Intuition Behind Chebyshev’s Inequality

In 2 recent Statistics Lessons of the Day, I introduced Markov’s inequality. explained the motivation and intuition behind Markov’s inequality. Chebyshev’s inequality is just a special version of Markov’s inequality; thus, their motivations and intuitions are similar. Markov’s inequality roughly says that a random variable is most frequently observed near its expected value, .  Remarkably, it quantifies just […]

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EM Algorithm for Bayesian Lasso R Cpp Code

September 5, 2014
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EM Algorithm for Bayesian Lasso R Cpp Code

Bayesian Lasso $$\begin{align*} p(Y_{o}|\beta,\phi)&=N(Y_{o}|1\alpha+X_{o}\beta,\phi^{-1} I_{n{o}})\\ \pi(\beta_{i}|\phi,\tau_{i}^{2})&=N(\beta_{i}|0, \phi^{-1}\tau_{i}^{2})\\ \pi(\tau_{i}^{2})&=Exp \left( \frac{\lambda}{2} \right)\\ \pi(\phi)&\propto \phi^{-1}\\ \pi(\alpha)&\propto 1\\ \end{align*}$$ Marginalizing over \(\alpha\) equates to centering the observations and losing a degree of freedom and working with the centered \( Y_{o} \). Mixing over \(\tau_{i}^{2}\) leads to a Laplace or Double Exponential prior on \(\beta_{i}\) with rate parameter \(\sqrt{\phi\lambda}\) […] The post EM Algorithm for Bayesian Lasso R Cpp Code appeared first on Lindons…

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Mathematical Statistics Lesson of the Day – Chebyshev’s Inequality

Mathematical Statistics Lesson of the Day – Chebyshev’s Inequality

The variance of a random variable is just an expected value of a function of .  Specifically, . Let’s substitute into Markov’s inequality and see what happens.  For convenience and without loss of generality, I will replace the constant with another constant, . Now, let’s substitute with , where is the standard deviation of . […]

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Monotonic Sequential Continuity

September 2, 2014
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This problem is the continuation of my previous post on Monotonic Sequence.ProblemProve the following: If $A_k$ is monotone, then \begin{equation} \mathrm{P}\left(\displaystyle\lim_{n\to\infty} A_n\right)=\displaystyle\lim_{n\to \infty}\mathrm{P}(A_n)....

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Monotonic Sequence

August 29, 2014
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Analysis with Programming has recently been accepted as a contributing blog on Mathblogging.org, a blogosphere aiming to be the best place to discover mathematical writing on the web. And as a first post, being a member of the said site, I will do prov...

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Mathematical and Applied Statistics Lesson of the Day – The Motivation and Intuition Behind Markov’s Inequality

Mathematical and Applied Statistics Lesson of the Day – The Motivation and Intuition Behind Markov’s Inequality

Markov’s inequality may seem like a rather arbitrary pair of mathematical expressions that are coincidentally related to each other by an inequality sign: where . However, there is a practical motivation behind Markov’s inequality, and it can be posed in the form of a simple question: How often is the random variable “far” away from […]

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