Boole’s Inequality

Booles' Inequality

Boole’s Inequality The Boole’s Inequality Theorem states that "the probability of several events occuring is less than or equal to the sum of the probabilities of each event occuring". For any two events $A$ and $B$, we have $$ \begin{eqnarray*} P(A \cup B) &=& P(A) + P(B) – P(A\cap B)\\ &\leq & P(A) + P(B)\\ …

Read more

Weibull Distribution Examples | Calculator | Two Parameter

Weibull Distribution Calculator with Examples

Weibull Distribution Calculator Use this calculator to find the probability density and cumulative probabilities for two parameter Weibull distribution with parameter $\alpha$ and $\beta$. Weibull Distribution Calculator Location parameter $\alpha$: Scale parameter $\beta$ Value of x Calculate Results Probability density : f(x) Probability X less than x: P(X < x) Probability X greater than x: …

Read more

Bonferroni Inequality

Bonferronis Inequality

Bonferroni’s Inequality For $n$ events $A_1,A_2,\cdots, A_n$ $$ \begin{equation}\label{bof} P\big(\cap_{i=1}^n A_i\big)\geq \sum_{i=1}^n P(A_i) -(n-1). \end{equation} $$ Proof For $n =2$, $$ \begin{equation*} P(A_1\cup A_2)= P(A_1)+P(A_2) -P(A_1\cap A_2) \end{equation*} $$ But $P(A_1\cup A_2)\leq 1$. Using this in above equation, we have $$ \begin{eqnarray*} & &P(A_1\cup A_2)= P(A_1)+P(A_2) -P(A_1\cap A_2)\leq 1 \\ \implies & & P(A_1\cap A_2) …

Read more