Statistics

Here is the list of few probabilistic results. Markov’s inequality If \(X\) is a nonnegative random variable and \(a>0\), then the probabilty that \(X\) is at least \(a\) is at most the expectation of \(X\) divided by \(a\): $$P(X\geq a) \leq \frac{E(X)}{a}.$$ Proof $$ \begin{aligned} E(X) &= \int_{-\infty}^\infty f(x)xdx = \int_0^\infty f(x)xdx\\ &= \int_0^a f(x)xdx + \int_a^\infty f(x)xdx\\ &\geq \int_a^\infty f(x)xdx \\ &\geq \int_a^\infty f(x)dx \\ &=a\int_a^\infty f(x)dx \\ &= aP(X\geq a), \end{aligned} $$ which, as you can see, is quite weak bound and proof itself is quite a troll....

The expected value, \(\mathbb E[x]\), is the weighted average of possible values of a random variable, with weights given by their respective theoretical probabilities. The variance, \(\mathbb D[X]\) (or \(\sigma^2,\text{Var}(X)\)), is the expectation of the squared deviation of a random variable from its population mean or sample mean. Bernoulli distribution The Bernoulli distribution essentially models a single trial of flipping a weighted coin. It is the probability distribution of a random variable taking on only two values, \(1\) and \(0\) with complementary probabilities \(p\) and \(1-p\) respectively....