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The relative frequency histogram is a visual representation of the probability distribution for a discrete random variable, such as the size of a pizza. It is similar to a typical bar chart, except that it shows the relative frequency (or proportion) of the data falling into each category.

To create this type of histogram, one would plot the sizes of the pizzas on the horizontal axis (the x-axis) and their corresponding proportions or relative frequencies on the vertical axis (the y-axis). Each pizza size category would have a bar (or rectangle) whose height represents the proportion of orders for that size. For example, in the problem statement, the 12-inch pizza would be represented by a bar with a height of 0.20 since 20% of the recent orders were for that size. Similarly, heights of 0.25, 0.50, and 0.05 would represent the 14-inch, 16-inch, and 18-inch pizzas respectively. This visual tool is invaluable for quickly understanding how the sizes of pizzas are distributed among the orders.

Cumulative probability refers to the probability that a random variable is less than or equal to a particular value. It gives us insight into the likelihood of a variable falling within a certain range up to a point. In this case, the exercise requires calculating the cumulative probability of pizza orders being less than 16 inches and less than or equal to 16 inches.

To calculate cumulative probability, you sum up the probabilities of all outcomes up to and including the one of interest. For example, to find \(P(x<16)\), one would add the probabilities of ordering a 12-inch pizza and a 14-inch pizza, which ultimately equals 0.45 or 45%. If one wants to calculate \(P(x \<= 16)\), one would include the probability of the 16-inch pizza in the sum as well, leading to a cumulative probability of 0.95, or 95%. This gives customers a comprehensive idea of the likelihood of ordering pizza sizes up to 16 inches.

The mean value approximation in a probability distribution is an estimate of the 'center' or 'average' outcome. It provides a single number that summarizes the central tendency of the data. In our pizza shop example, the mean size of an ordered pizza is given as approximately 14.8 inches.

One common question is: what is the probability that an observation is within a certain range around this mean? In the provided exercise, the task is to approximate the probability that the size of a pizza \(x\) is within 2 inches of this mean value, thus between 12.8 inches and 16.8 inches. Since only 14 inches and 16 inches fall within this range, we add their probabilities, resulting in 0.75 or 75%. This method of approximation gives us a way to understand variability relative to the mean, offering practical insights into the spread of the data around the average value.