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Chapter 8: Problem 62

How wide should the bars in a histogram be so that the probability \(P(a \leq X \leq b)\) equals the area of the corresponding portion of the histogram?

Short Answer

Expert verified
To ensure that the probability \(P(a \leq X \leq b)\) equals the area of the corresponding portion of the histogram, the width of the bars (h) should be chosen so that it closely approximates the PDF. Calculate it by dividing the range of the data by the number of bars, \(k\): Width of a bar (h) = \(\frac{max\{X\} - min\{X\}}{k}\) The choice of the number of bars and the width of the bars will depend on the specific distribution and the available data points. In general, a larger number of bars (i.e., smaller width) will generate a better approximation.

Step by step solution

01

Define the probability density function (PDF) and its properties

Recall that the probability density function is a function that describes the relative likelihood of a continuous random variable taking a particular value. The area under the curve of the PDF over a certain interval \((a, b)\) represents the probability that the random variable will take a value within that interval. The main properties we need for this problem are the mean, the variance, and the range of the random variable. Each of these properties helps to define the width of the bars in the histogram.
02

Histogram approximation of the PDF

A histogram is an approximate representation of the PDF and consists of a number of bars whose height is determined by the frequency of the interval represented by the bar. The more bars we have, the better the histogram approximates the PDF. To determine the width of the bars, we need to consider the total area of the histogram (which should equal 1) and the number of bars.
03

Divide the range by the number of bars

Divide the range of the data (i.e., the difference between the maximum and minimum values) by the number of bars we want to use in the histogram, let's say \(k\). This will give us the width of each bar. Width of a bar (h) = \(\frac{max\{X\} - min\{X\}}{k}\)
04

Find the area of each bar

The area of each bar in the histogram is given by the product of its width and height, i.e., Area of a bar = Height of the bar 脳 Width of the bar In our case, the width of the bar is given in step 3.
05

Determine the probability using the area of each bar

To find the probability \(P(a \leq X \leq b)\), now we sum up the areas of all the bars in the histogram that lie within the interval \((a, b)\). This total area will be equal to the probability we are looking for. To ensure that the probability is equal to the area under the histogram over a specified interval, the width of the bars in the histogram should be chosen such that it closely approximates the PDF. In general, a larger number of bars (i.e., smaller width) will generate a better approximation. However, the choice of the number of bars and the width of the bars will depend on the specific distribution and the available data points.

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