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 variable the 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 variable is 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 , and the proportion of
the population with measurements between two numbers and is
the area under the curve and between and , as shown in Figure
2.6 "A Very Fine Relative Frequency Histogram" in Chapter 2
"Descriptive Statistics". If we think of as a measurement to
infinite precision arising from the selection of any one member of the
population at random, then is simply the
proportion of the population with measurements between and ,
the curve in the relative frequency histogram is the density function
for , and we arrive at the definition just above.
Every density function must satisfy the following two conditions:
- For all numbers , so that the graph of never drops below the -axis.
- The area of the region under the graph of and above the -axis is .
Because the area of a line
segment is , the definition of the probability distribution of a
continuous random variable implies that for any particular decimal
number, say , the probability that assumes the exact value
is . This property implies that whether or not the endpoints
of an interval are included makes no difference concerning the
probability of the interval.
A random variable has 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 to and is the -axis everywhere else. Since the total area under the curve must be , the height of the horizontal line is . See Figure 5.4 "Probability of Waiting At Most Minutes for a Bus". The probability sought is . By definition, this probability is the area of the rectangular region bounded above by the horizontal line , bounded below by the -axis, bounded on the left by the vertical line at (the -axis), and bounded on the right by the vertical line at . This is the shaded region in Figure 5.4 "Probability of Waiting At Most Minutes for a Bus". Its area is the base of the rectangle times its height, . Thus .
Figure 5.4 Probability of Waiting At Most Minutes for a Bus
Most people have heard of the "bell curve". It is the graph of a specific density function that 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
where and is the base of the natural logarithms.
Each different choice of specific numerical values for the pair and gives a different bell curve. The value of determines the location of the curve, as shown in Figure 5:5 "Bell Curves with ". In each case the curve is symmetric about .
Figure 5.5 Bell Curves with and Different Values of
The value of determines whether the bell curve is tall and thin or short and squat, subject always to the condition that the total area under the curve be equal to . This is shown in Figure 5.6 "Bell Curves with ", where we have arbitrarily chosen to center the curves at .
Figure 5.6 Bell Curves with and Different Values of
The probability distribution corresponding to the density function for the bell curve with parameters and is called the normal distribution with mean and standard deviation .
A continuous random variable whose probabilities are described by the normal distribution with mean and 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 and standard deviation . We repeat an important fact about this curve:
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 and Standard Deviation
Heights of 25 -year-old men in a certain region have mean inches 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, , and is symmetric about the mean.
Figure 5.8 Density Function for Heights of 25-Year-Old Men
Since the total area under the curve is , by symmetry the area to the right of is half the total, or . But this area is precisely the probability , the probability that a randomly selected 25 year-old man is more than inches tall.
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.