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 . […]
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 […]
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 […]
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 . […]
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 […]
Markov’s inequality is an elegant and very useful inequality that relates the probability of an event concerning a non-negative random variable, , with the expected value of . It states that where . I find Markov’s inequality to be beautiful for 2 reasons: It applies to both continuous and discrete random variables. It applies to any non-negative […]
In math and science, we learn the equation of a line as , with being called the dependent variable and being called the independent variable. This terminology holds true for more complicated functions with multiple variables, such as in polynomial regression. I highly discourage the use of “independent” and “dependent” in the context of statistics […]
I first started my video tutorial series on survival analysis by defining the hazard function. I then explained how this definition leads to the elegant relationship of . In my new video, I derive 6 useful mathematical relationships that exist between any 2 of the 3 quantities in the above equation. Each relationship allows one quantity […]
According to the central limit theorem, if random variables, , are independent and identically distributed, is sufficiently large, then the distribution of their sample mean, , is approximately normal, and this approximation is better as increases. One of the most remarkable aspects of the central limit theorem (CLT) is its validity for any parent distribution of […]
Having taught and tutored introductory statistics numerous times, I often hear students misinterpret the Central Limit Theorem by saying that, as the sample size gets bigger, the distribution of the data approaches a normal distribution. This is not true. If your data come from a non-normal distribution, their distribution stays the same regardless of the […]
