Category: Math

What is the intuitive explanation for the duality in optimization?

Ref: Let us a consider the problem minimizef(x) subject to g(x)≤0minimizef(x) subject to g(x)≤0 where f and g are convex.  Let x∗x∗ denote the constrained minimizer and p∗=f(x∗)p∗=f(x∗). The dual function is defined as h(λ)=minxf(x)+λg(x)h(λ)=minxf(x)+λg(x), and given appropriate conditions, we have maxλh(λ)=p∗maxλh(λ)=p∗.  Why is that? Let's plot all the possible values that (g(x), f(x)) can take for every point … Continue reading What is the intuitive explanation for the duality in optimization?

Sample Mean Variance

Ref: Let X1, X2, ... , Xn  be a random sample of size n from a distribution (population) with mean μ and variance σ2. What is the variance of X¯? Var(X¯) named sample mean variance. Solution. Starting with the definition of the sample mean, we have: Var(X¯)=Var(X1+X2+⋯+Xnn)Var(X¯)=Var(X1+X2+⋯+Xnn) Rewriting the term on the right so that it is clear that we have a linear combination of … Continue reading Sample Mean Variance

Confidence Interval of Normal Distribution

Ref: Confidence intervals are a little bit tricky in a sense that people don't define what they really mean by confidence interval. Now let me tell you a scenario using which you can start understanding CIs on a very basic level. Imagine you want to find the mean height of all the people in a … Continue reading Confidence Interval of Normal Distribution

Intuitive Meaning of Covariance

Ref: Sometimes we can "augment knowledge" with an unusual or different approach. I would like this reply to be accessible to kindergartners and also have some fun, so everybody get out your crayons! Given paired (x,y)(x,y) data, draw their scatterplot. (The younger students may need a teacher to produce this for them. 🙂 Each pair … Continue reading Intuitive Meaning of Covariance