For discrete random variables, one simple procedürê iš tõ list the probabilities of all pos- sible outcomes according to the values of x. Probability Distribution Function The probability distribution function, P(x), of a discrete random variable X represents the probability that X takes the value x, as a function of x. That is, P(x) = P(X = x), for all values of x We use the term probability distribution to represent probability distribution functions in this book, following the common practice. Once the probabilities have been calculated, the probability distribution function can be graphed. Example 4.1 Number of Product Sales (Probability Distribution Graph) Define and graph the probability distribution function for the number of waffles sold by a bakery in the Netherlands. This shop offers waffles that have a price of €4.00 each. Solution Let the random variable X denote the number of sales during a single hour of business from 3 to 5 P.M. The probability distribution of sales is given by Table 4.1, and Figure 4.1 is a graphical picture of the distribution. Table 4.1 Probability Distribution for Example 4.1 P(x) 0.10 0.30 2 0.20 3 0.40 Figure 4.1 Graph of Probability Distribution for Example 4.1 Probability Distribution for Waffle Sales 0.4 - 0.40 0.30 0.3 - 0.2 0.20 0.10 0.1 0.0 x (Number of Waffles Sold) From the probability distribution function, we see that, for example, the prob- ability of selling one waffle is 0.30 and the probability of selling two or more 0.60(0.20 + 0.40 ). 4.2 Probability Distributions for Discrete Random Variables 153

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Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.  

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.