CHAPTER Probability Density Functions

Transcription

1 CHAPTER 8 1. Probability Density Functions For some types of data, we want to consider the distribution throughout the population. For example, age distribution in the US gives the percentage of the US population that is in a given age range. One way to represent data of this sort is with a special type of histogram. Recall that a histogram is a graph showing the frequency of some quantity over an interval using rectangles. The interval is divided into sections, called bins. Frequency is on the y-axis, while the interval is on the x-axis. Histograms are not bar graphs; bar graphs show frequencies for categorical data, histograms are used for continuous data. For data showing the distribution, we will create a histogram so that the area of each rectangle represents the percentage of the population in that bin. Consider the following data: Age Group % of total pop. Width Height 2 29% % % % 8 1 3% Example 1. Using the histogram, what percentage of the population is between 2 and 6 years old? What percentage of the population is older than 9? 1

2 2 CHAPTER 8 Notice that in order to find the percentage of the population, we consider the area underneath the rectangles. We could get better information about the age distribution if we had smaller bins. As the bins get very small, the top of the histogram begins to look like a smooth continuous function. If we wanted to determine the percentage of the population between and 15 years old, we still look at the area underneath the distribution. As bin size goes to zero, however, this is no longer involves the areas of the rectangles. Instead, we use the integral. Suppose the age density function the function created by the top of the histogram as the bin size goes to is given by p(t). The units of p(t) are percent per year; we rarely use the density function itself, since we are usually far more interested in the percentage of the population. The fraction of the population that is between and 15 years old is the area under p(t) between t = and t = 15, or 15 p(t)dt. In general, the fraction of the population that is between a and b years old is b a p(t)dt. One of the most important aspects of the density function is the area under the curve between the lowest and highest values of t. 1 p(t)dt represents the fraction of the population between and 1 (Our data is slightly flawed because it doesn t account for people over 1, but they make up a miniscule percentage of the population). So, assuming that every person is between and 1, 1 p(t)dt = 1. Definition 1. A function, p(x) is a density function if The fraction of the population for which x is between a and b is b a p(x)dx p(x)dx = 1 p(x) for all x. Notice that in the second criterion, the bounds of the integral are ±. Some intervals, like age, are finite, so the improper integral will be the same as 1 p(t)dt. Other intervals

3 CHAPTER 8 3 can be infinite. When you are checking p(x)dx = 1 for a density function on a finite interval, you only need to integrate between the smallest and largest possible values.

4 4 CHAPTER 8 2. Cumulative Distribution Functions Given a density function, p(x), we can define a cumulative density function, P (x). Cumulative density functions are really just a shorthand way to talk about finding the percentage of the population between x = a and x = b. Definition 2. A cumulative density function P (t) of a density function p(x) is defined by P (t) = t p(x)dx. So, P (t) is the fraction of the population having values of x below t. Note that as before, if x can t be less than some number a, it is equivalent to compute t a p(x)dx. Notice that by the Second Fundamental Theorem of Calculus, P (t) = p(t). P (t) has the following properties: P is increasing or nondecreasing. lim t P (t) = 1 and lim t P (t) =. The fraction of the population having values of x between a and b is b a p(x)dx = P (b) P (a) Usually, we use cumulative density functions to talk about probability. Since density functions are used on intervals, cumulative density functions are used to describe continuous variables. Remember that in probability, a continuous variable is one that can take on any value in an interval. For example, age is a continuous variable because any age between and 1 is possible, like or Variables that aren t continuous are discrete; the color of a randomly chosen m&m is a discrete variable, because there are only 6 options. If P (t) is a cumulative density function describing the probability of some event x, P (t) is the probability that x < t. So, if p(x) is the age density function we discussed before, P (t) = t p(x)dx is the probability that a randomly chosen person is younger than t years old. This lines up with our old interpretation of t p(x)dx as the fraction of the population younger than t. Example 2. p(x) =.2.2x, x 1 is a density function describing the amount of time x, in minutes, a person waits while on hold with Verizon. Find and graph the cumulative density function and find the following: P()

5 The probability of waiting less than 2 minutes The probability of waiting between 4 and 6 minutes The probability of waiting more than 5 minutes The probability of waiting more than 1 minutes CHAPTER 8 5 CAUTION P (a) is NOT the probability that x = a. It is the probability that x < a. For continuous variables, the probability that x is exactly a is actually zero, because there are infinite possible values for x. Remember that for a random variable (the event we want to find the probability of), the maximum probability is 1. Something always has to happen, so the probability of a certain event not happening is 1 minus the probability of that event happening. Since P (a) is the probability of x < a, The probability that x > a is 1 P (a).

6 6 CHAPTER 8 3. Median and Mean We can define the median and the mean of a distribution, just like we do for other data Median. Given an ordered set of discrete numbers, like 36, 42, 47, 47, 59, 7, 73, 89, 157, the median is the middle value. In this case, the median is 59. When we have a quantity x distributed through a population, the median is the value T so that half the population has values of x less than or equal to T and half the population has values of x greater than or equal to T. If p(x) is the density function, this means that half the area under the graph of p(x) lies to the left of T, or or, equivalently, P (T ) =.5. T p(x)dx =.5 Example 3. Given the density function p(x) =.2.2x, x 1 find the median Mean. To understand the formula for the mean, let s return to the age density function, p(x). If we want to find the average age of people in the US, intuitively we should add together the ages of everyone in the US and divide by the total number of people living in the US. This, of course, is impossible. Instead, we can try the following: First, let N be the total number of people living in the US. Then the number of people who are t years old is N times the percentage of people who are t years old. We can t use p(x) to find the percent of people who are exactly t years old. Instead, we can find the percent of people whose age is between t and t + t, where t is some small number. The percentage of the population with age between t and t + t is p(t) t. Then the number of people with age between t and t + t is N p(t) t. The last piece of information we need is the sum of all of their ages. All of the ages are very close to t, so the sum of the ages is approximately t N p(t) t.

7 CHAPTER 8 7 Now, we need to add up these estimates over all the intervals, giving us an estimate of the sum of ages of all people in the US: tp(t) tn This estimate gets better as t gets smaller. As t goes to, the sum becomes 1 Ntp(t)dt = N 1 tp(t)dt The mean is the sum of all ages divided by the number of people in the US, N. Thus, the mean is N 1 tp(t)dt = N 1 tp(t)dt In general, if a quantity has density function p(x), the mean value of the quantity is xp(x)dx Example 4. Given the density function p(x) =.2.2x, x 1, find the mean The Normal Distribution. The normal density function is used for many different quantities, from height to SAT scores. We describe normal distributions by the mean µ and the standard deviation σ. The standard deviation measures how spread out the normal distribution is. The normal distribution has a density function of the form p(x) = 1 σ (x µ) 2 2π e 2σ 2. Example 5. The number of eggs in a Bufo americanus egg mass can be modeled by a normal distribution with mean 8, and standard deviation 1. Write down the density function and find the probability that an egg mass contains less than 1,5 eggs.

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