Part a
Calculate \(P(X=k)\) for a binomial random variable when \(k = 0\) and when \(k = n\). Use the binomial probability formula \(P(X=k)= \binom{n}{k} p^k (1-p)^{n-k}\). Simplify your answers.
Optimisation of a Piecewise Defined Function
A company has a pricing model given by a piecewise function for its service charge, defined as follows: \(C(x) = \begin{cases} 30 – 2x & \text{if } 0 \leq x \leq 10, \\ 10 & \text{if } 10 20 \end{cases}\) where \(x\) represents the amount of service units sold and \(C(x)\) the cost in dollars. […]
Optimisation of a Piecewise-Defined Function on a Restricted Domain
Consider the piecewise function \(f(x)\) defined by: \[ f(x) = \begin{cases} x^2 – 4 & \text{if } x < 2, \\ 8 – x & \text{if } x \geq 2 \end{cases} \] and let the domain be \([-3, 5]\). Determine the absolute minimum and absolute maximum values of \(f(x)\) on the given interval. Clearly state […]
Analyzing Partial Merges in Voronoi Diagrams
Consider a configuration of seed points in the plane. In constructing a Voronoi diagram, under what conditions might two or more cells exhibit a partially merged boundary? Discuss how such partial merges arise and explain the implications they have on defining the precise edges of the cells.