Math 2015

Lesson 20

Probability and Cumulative Distribution Functions Last time, we discussed density functions and probability. Today, we continue our discussion and begin to talk about cumulative distribution functions.

Reminders and Practice: Density Functions and Probability If you have a density function p(x) for a characteristic x of a population, you can find out what fraction of the population have an x between a and b by integrating p: ⎛ fraction of the ⎞ ∫a p(x)dx = ⎜⎜ population for ⎟⎟ ⎝ which a ≤ x ≤ b⎠ b

We determined that the integral ∫−∞ p(x)dx = 1, where we will interpret the infinity symbol to just mean “as far as we can go.” ∞

We also determined that we can interpret density functions in terms of probabilities: If we have a probability density function p(x) for a random variable, we can determine the probability of x being between a and b using ⎛ probability⎞ ∫a p(x)dx = ⎜⎜ that ⎟⎟ ⎝ a≤ x≤b ⎠ b

Let’s get in a little more practice interpreting these values. Example:

Suppose the density function below is for the grades of all 2015 students on Test 1 last semester. 0.03 0.025 0.02 0.015 0.01 0.005 20

40

60

80

100

According to this density function, • What is the most common letter grade from Test 1? (Ignore +/-) • Make a rough estimate of the probability that a randomly selected student would have earned an A.

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Math 2015

Lesson 20

Cumulative Distribution Functions Suppose we have a density function p(x) for some quantity. We know that the integral from a to b of p(x) gives the probability that x is between a and b. We define the cumulative distribution function for the quantity to be

In terms of populations, this will be the proportion of the population for which the quantity measured is less than x. In terms of probabilities, this means the probability that the quantity is less than x. Example: Last class, we discussed a fair spinner, which could land at any angle between 0 and 360 degrees. We determined a density function for the angle was 1 for x between 0 and 360, and zero elsewhere. The cumulative distribution p(x) = 360

function for this spinner is

The probability steadily increases that the spinner will land at an angle less than x as x increases from zero (where P is 0) to 360 (where P is 1). The density function and cumulative distribution functions are graphed below. Note how the cumulative distribution function “accumulates” the probabilities shown in the density function until it reaches 1.

1

1 360 360

Density Function

360

Cumulative Distribution Function

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Math 2015

Lesson 20

Properties of Cumulative Distribution Functions Some of the properties that we noticed about the cumulative distribution function for the spinner are general properties of all cumulative distribution functions. For example: • If P(x) is a cumulative distribution function for a quantity, then P(x) is the probability of this quantity being less than x. (Example: If P(x) is a cumulative distribution function for the age (in months) of fish in a lake, then P(10) is the probability that a randomly selected fish is 10 months old or younger.) • As x gets smaller, P(x) must approach zero. We can express this by saying

This says that as you make x smaller, the probability of getting a value less than x gets closer to 0. In our examples, we often have p(x) > 0 only on a finite interval. In these cases, if we choose x small enough, then P(x) = 0. (This is what happened with the spinner; only values between 0 and 360 were permitted.) •

As x gets bigger, P(x) must approach one. We can write

•

This says that as you make x bigger, the probability of getting a value less than x gets closer to one. In other words, as more of the possibilities are taken into account, we get closer to 1. (Something must happen.) P(x) is always We can see this is true because the derivative is p, and p(x) cannot be negative. If we think about probabilities, it also makes sense: the probability of having a result less than 4 cannot be smaller than the probability of having a result less than 3, for example.

Practice With CDFs Example:

Suppose we have determined that the life expectancy of a certain electronic component in days is well represented by the density function 1 p(x) = 2 if x ≥ 1, and p(x) = 0 for x < 1. x a) Find the probability that the component lasts between 0 and 1 day.

b) Find the probability that the component lasts from 0 to 10 days. 104

Math 2015

Lesson 20

c) Find the probability that the component lasts more than 10 days.

d) Find the cumulative distribution function for the life expectancy of the component.

Example:

Now suppose someone said that the following graph for the cumulative distribution function for the distributions of grades on the 2015 final: 1

0.5

25

50

75

100

If the above is the cumulative distribution function, what is the probability that a random student scored • 25 or lower? • 50 or lower? What must we conclude?

Relating the CDF and DF The cumulative distribution function and the density function are related by the Fundamental Theorem of Calculus (Version II). Since ,

we know that In other words, p(x) is the rate of change of the cumulative distribution function. Or equivalently: P(x) is an antiderivative for p(x).

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Math 2015 Example:

Lesson 20 Suppose the cumulative distribution function is as shown below: 2 1.5 1 0.5 -2

-1

1

2

3

-0.5 -1

Sketch the density function:

When we work with density functions and cumulative distribution functions, it is vital that we know which is which, since we answer questions about probability in different ways with density functions than we do with cumulative distribution functions.

Summary Today, we have •

Defined a cumulative distribution function P in terms of a density function p:

P(x) =

∫

x −∞

p(t) dt

•

Determined properties of a cumulative distribution function, such as the fact that it increases towards 1 as x increases.

•

Related the cumulative distribution function to the density function by noting that since P(x) =

∫

x −∞

p(t) dt , P ′(x) = p(x) .

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