Continuous Random Variables
First, this section talks about how to describe continuous distributions and compute related probabilities, including some basic facts about the normal distribution. Then, it covers how to compute probabilities related to any normal random variable and gives examples of using-score transformations. Finally, it defines tail probabilities and illustrates how to find them.
Continuous Random Variables
- To learn the concept of the probability distribution of a continuous random variable, and how it is used to compute probabilities.
- To learn basic facts about the family of normally distributed random variables.
The Probability Distribution of a Continuous Random Variable
For a discrete random variablethe probability that assumes one of its possible values on a single trial of the experiment makes good sense. This is not the case for a continuous random variable. For example, suppose denotes the length of time a commuter just arriving at a bus stop has to wait for the next bus. If buses run every minutes without fail, then the set of possible values of is the interval denoted , the set of all decimal numbers between and . But although the number is a possible value of , there is little or no meaning to the concept of the probability that the commuter will wait precisely minutes for the next bus. If anything the probability should be zero, since if we could meaningfully measure the waiting time to the nearest millionth of a minute it is practically inconceivable that we would ever get exactly minutes. More meaningful questions are those of the form: What is the probability that the commuter's waiting time is less than minutes, or is between and minutes? In other words, with continuous random variables one is concerned not with the event that the variable assumes a single particular value, but with the event that the random variable assumes a value in a particular interval.
The probability distribution of a continuous random variableis an assignment of probabilities to intervals of decimal numbers using a function , called density function, in the following way: the probability that assumes a value in the interval is equal to the area of the region that is bounded above by the graph of the equation , bounded below by the -axis, and bounded on the left and right by the vertical lines through and , as illustrated in Figure 5.1 "Probability Given as Area of a Region under a Curve".
Figure 5.1 Probability Given as Area of a Region under a Curve
This definition can be
understood as a natural outgrowth of the discussion in Section 2.1.3
"Relative Frequency Histograms" in Chapter 2 "Descriptive Statistics".
There we saw that if we have in view a population (or a very large
sample) and make measurements with greater and greater precision, then
as the bars in the relative frequency histogram become exceedingly fine
their vertical sides merge and disappear, and what is left is just the
curve formed by their tops, as shown in Figure 2.5 "Sample Size and
Relative Frequency Histograms" in Chapter 2 "Descriptive Statistics".
Moreover the total area under the curve is
Every density function
- For all numbers , so that the graph of never drops below the -axis.
- The area of the region under the graph of
Because the area of a line
For any continuous random variable:
A random variablehas the uniform distribution on the interval : the density function is if is between and and for all other values of , as shown in Figure 5.2 "Uniform Distribution on".
Figure 5.2 Uniform Distribution on
- Find , the probability that assumes value greater than .
- Find , the probability that assumes value less than or equal to .
- Find , the probability that assumes value between and .
- is the area of the rectangle of height and base length , hence is . See Figure 5.3 "Probabilities from the Uniform Distribution on "(a).
- is the area of the rectangle of height and base length , hence is . See Figure 5.3 "Probabilities from the Uniform Distribution on "(b).
- is the area of the rectangle of height and length , hence is . See Figure 5.3 "Probabilities from the Uniform Distribution on "(c).
Figure 5.3 Probabilities from the Uniform Distribution on
A man arrives at a bus stop at a random time (that is, with no regard for the scheduled service) to catch the next bus. Buses run every 30 minutes without fail, hence the next bus will come any time during the next 30 minutes with evenly distributed probability (a uniform distribution). Find the probability that a bus will come within the next 10 minutes.
The graph of the density function is a horizontal line above the interval from
Figure 5.4 Probability of Waiting At Most
Most people have heard of the "bell curve". It is the graph of a specific density functionthat describes the behavior of continuous random variables as different as the heights of human beings, the amount of a product in a container that was filled by a high-speed packing machine, or the velocities of molecules in a gas. The formula for contains two parameters and that can be assigned any specific numerical values, so long as is positive. We will not need to know the formula for , but for those who are interested it is
Each different choice of specific numerical values for the pair
Figure 5.5 Bell Curves with
The value of
Figure 5.6 Bell Curves with
The probability distribution corresponding to the density function for the bell curve with parametersand is called the normal distribution with mean and standard deviation .
A continuous random variable whose probabilities are described by the normal distribution with meanand standard deviation is called a normally distributed random variable, or a normal random variable for short, with mean and standard deviation .
Figure 5.7 "Density Function for a Normally Distributed Random Variable with Mean" shows the density function that determines the normal distribution with mean
The density curve for the normal distribution is symmetric about the mean.
Figure 5.7 Density Function for a Normally Distributed Random Variable with Mean
Heights of 25 -year-old men in a certain region have meaninches and standard deviation inches. These heights are approximately normally distributed. Thus the height of a randomly selected 25 -year-old man is a normal random variable with mean and standard deviation . Sketch a qualitatively accurate graph of the density function for . Find the probability that a randomly selected 25-year-old man is more than inches tall.
The distribution of heights looks like the bell curve in Figure 5.8 "Density Function for Heights of 25 Year-Old Men". The important point is that it is centered at its mean,
Figure 5.8 Density Function for Heights of 25-Year-Old Men
Since the total area under the curve is
We will learn how to compute other probabilities in the next two sections.
This text was adapted by Saylor Academy under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License without attribution as requested by the work's original creator or licensor.