Probability and Cumulative Distribution Functions

Transcription

1 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() for a characteristic of a population, you can find out what fraction of the population have an between a and b by integrating p: a b fraction of the p()d = population for which a b We determined that the integral p()d =, 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() for a random variable, we can determine the probability of being between a and b using a b probability p()d = that a b Let s get in a little more practice interpreting these values. Suppose the density function below is for the grades of all 205 students on Test last semester According to this density function, What is the most common letter grade from Test? (Ignore +/-) Make a rough estimate of the probability that a randomly selected student would have earned an A. 02

2 Cumulative Distribution Functions Suppose we have a density function p() for some quantity. We know that the integral from a to b of p() gives the probability that 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. In terms of probabilities, this means the probability that the quantity is less than. Last class, we discussed a fair spinner, which could land at any angle between 0 and degrees. We determined a density function for the angle was p() = for between 0 and, and zero elsewhere. The cumulative distribution function for this spinner is The probability steadily increases that the spinner will land at an angle less than as increases from zero (where P is 0) to (where P is ). 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. Density Function Cumulative Distribution Function 03

3 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 eample: If P() is a cumulative distribution function for a quantity, then P() is the probability of this quantity being less than. ( If P() is a cumulative distribution function for the age (in months) of fish in a lake, then P(0) is the probability that a randomly selected fish is 0 months old or younger.) As gets smaller, P() must approach zero. We can epress this by saying This says that as you make smaller, the probability of getting a value less than gets closer to 0. In our eamples, we often have p() > 0 only on a finite interval. In these cases, if we choose small enough, then P() = 0. (This is what happened with the spinner; only values between 0 and were permitted.) As gets bigger, P() must approach one. We can write This says that as you make bigger, the probability of getting a value less than gets closer to one. In other words, as more of the possibilities are taken into account, we get closer to. (Something must happen.) P() is always We can see this is true because the derivative is p, and p() 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 eample. Practice With CDFs Suppose we have determined that the life epectancy of a certain electronic component in days is well represented by the density function p() = 2 if, and p() = 0 for <. a) Find the probability that the component lasts between 0 and day. b) Find the probability that the component lasts from 0 to 0 days. 04

4 c) Find the probability that the component lasts more than 0 days. d) Find the cumulative distribution function for the life epectancy of the component. Now suppose someone said that the following graph for the cumulative distribution function for the distributions of grades on the 205 final: 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() is the rate of change of the cumulative distribution function. Or equivalently: P() is an antiderivative for p(). 05

5 Suppose the cumulative distribution function is as shown below: 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() = p(t) dt Determined properties of a cumulative distribution function, such as the fact that it increases towards as increases. Related the cumulative distribution function to the density function by noting that since P() = p(t) dt, P () = p(). 06