Hyper-geometric Distribution
Posted: October 21, 2021 #StatsThe hyper-geometric distribution is about the un-ordered sampling without replacement. Read more
Estimating Parameters - Maximum Likelihood and Moments
Posted: October 10, 2021 #StatsIf a phenomenon is likely to be described by a kind of distribution function, we might want to know the best parameters for the distribution function. There’re two ways to estimate the parameters based on a collection of samples, the method of maximum likelihood and the method of moments. Read m...
The Gamma Distribution
Posted: October 5, 2021 #StatsFirst of all, we define what gamma function is, then we proceed with gamma distribution, and followed by the discussion of its properties. Read more
The Negative Binomial Distribution
Posted: September 29, 2021 #StatsFollow the logic of geometric distribution, if we want to study the probability of $r^{\text{th}}$ success in $k^{\text{th}}$ of a series of trials, it must be the case that $(r −1)$ success occur during the first $(k −1)$ trials and the $r^{\text{th}}$ happens on exactly the $k^{\text{th}}$ tria...
The Geometric Distribution
Posted: September 28, 2021 #StatsConsider a series of independent trials and each has one of two outcomes, success or failure. If $p$ is the probability of success, the geometric distribution means the probability at which the first success occurs. This is the geometric distribution. Read more
A Weak Proof of Central Limit Theorem
Posted: September 21, 2021 #StatsCentral Limit Theorem (CLT) is one of the two most important theorems in statistics (the other is the Large Number Theorem). In this note, the theorem of moment generating function is used to prove the CLT. This is a weak proof, because the underlying logic is that by showing two varibles have th...
Taylor Theorem and its Proof
Posted: September 18, 2021 #Math123Taylor’s Theorem is a very powerful tool to approximate any functions that are infinitely differentiable on a certain interval between a and b. Of course, the exact value of a and b need to be carefully defined, so the formula/series developed by the theorem shall converge within the defined inte...
How to Deduce Stirling Formula
Posted: July 4, 2021 #Math123Stirling formula is the approximation to $n!$ This is a formula widely used in statistics theorem proofs. However, the detailed proof are often omitted in most textbooks. I digged into the online resources. Among numerous articles and papers, I referred to Marton Balazs and Balint Toth’s “Stirlin...
Poisson Distribution
Posted: June 30, 2021 #StatsIn the binomial distribution where n is quite large, it’s usually a tedious job to calculate k! when computer was not available back in the 18th to early 20th century. So Simeon Denis Poisson, a French mathematician came up with a approximation, which proves to be working quite well with a small ...
Moment Generating Function
Posted: June 25, 2021 #StatsMoment generating function is the expected value of $e^{tx}$ with respect to the pdf of the variable. If we differentiate mgf and then let $t=0$, we can get $E(X), E(X^2), \dots$. So mgf is a way to find $\mu$ and $\sigma$ if the pdf itself is complicated. Furthermore, mgf can also be used to pro...