Density Curves for a Continuous Random Variable

Density Curves

Previously we discussed discrete random variables, and now we consider the contuous type. A continuous random variable is such that all values (to any number of decimal places) within some interval are possible outcomes. A continuous random variable has an infinite number of possible values so we can't assign probabilities to each specific value. If we did, the total probability would be infinite, rather than 1, as it is supposed to be

To describe probabilities for a continuous random variable, we use a probability density function. A probability density function is a curve such that the area under the curve within any interval of values along the horizontal gives the probability for that interval.

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Normal Random Variables

The most commonly encountered type of continuous random variable is a normal random variable , which has a symmetric bell-shaped density function. The center point of the distribution is the mean value, denoted by μ (pronounced "mew"). The spread of the distribution is determined by the variance, denoted by σ² (pronounced "sigma squared") or by the square root of the variance called standard deviation, denoted by σ (pronounced "sigma").

Example : Suppose vehicle speeds at a highway location have a normal distribution with mean μ = 65 mph and standard deviation s = 5 mph. The probability density function is shown below. Notice that the horizontal axis shows speeds and the bell is centered at the mean (65 mph).

Probability for an Interval = Area under the density curve in that interval

The next figure shows the probability that the speed of a randomly selected vehicle will be between 60 and 73 mile per hour, with this probability equal to the area under the curve between 60 and 73.

Empirical Rule Review

Recall that our first lesson we learned that for bell-shaped data, about 95% of the data values will be in the interval mean ± (2 × std. dev) . In our example, this is 65 ± (2 × 5), or 55 to 75. The next figure shows that the probability is about 0.95 (about 95%) that a randomly selected vehicle speed is between 55 and 75.

The Empirical Rule also stated that about 99.7% (nearly all) of a bell-shaped dataset will be in the interval mean ± (3 × std. dev) . This is 65 ± (3 × 5), or 50 to 80 for example. Notice that this interval roughly gives the complete range of the density curve shown above.

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Finding Probabilities for a Normal Random Variable

Remember that the cumulative probability for a value is the probability less than or equal to that value. Minitab, SPSS, Excel, and the TI-83 series of calculators will give the cumulative probability for any value of interest in a specific normal curve.

For our example of vehicle speeds, here is Minitab output showing that the probability = 0.9452 that the speed of a randomly selected vehicle is less than or equal to 73 mph.

We can find this probability using either Minitab or SPSS:

• Using Minitab
• Using SPSS

To calculate normal random variable probabilities in Minitab:

1. Open Minitab without data.
2. From the menu bar select Calc>Probability Distribution> Normal.
3. Select the radio button for Cumulative Probability (this is the default option)
4. In the text box for Mean enter 65
5. In the text box for Standard Deviation enter 5
6. Since we do not have a column of data select the radio button for Input Constant and enter 73
7. Click OK
8. The output is as follows:

To calculate normal random variable probabilities in SPSS:

1. Open SPSS without data.
2. Because SPSS will not let you do anything without data just type something into the first blank cell (e.g. enter the number 2 in the first cell in column 1) and be sure to then click any other cell. You need to do this to complete the entry of the value into that cell.
3. From the menu bar select Transorm > Compute Variable
4. In the box for Target Variable enter any name (e.g. prob).
5. Click inside the box for Numeric Expression (this should put the cursor inside this box)
6. From the drop down menu for Function Group select CDF and Noncentral CDF
7. From the list of Functions and Special Variables select CDF.Normal
8. Click on the arrow next to the Delete button. This should put in the expression window the following: CDF.NORMAL(?,?,?)
9. Replace ? with 73,65,5 These values represent the observed value (73), the mean (65) and standard deviation (5). BE SURE TO INCLUDE THE COMMAS AND KEEP THE PARENTHESES!!
10. Click OK
11. In the Data worksheet you should see a column with the target label (e.g. prob) with the value 0.95 (If you click this cell and paste it into another document e.g. windows or notepad you will see the value is 0.945200708300442 but the number is rounded to two decimals in SPSS worksheet.)
12. If you used the above labeling the worksheet would look as follows:

Here is a figure that illustrates the cumulative probability we found using this procedure.

"Greater than" Probabilities

Sometimes we want to know the probability that a variable has a value greater than some value. For instance, we might want to know the probability that a randomly selected vehicle speed is greater than 73 mph, written P(X > 73).

For our example, probability speed is greater than 73 = 1 - 0.9452 = 0.0548.

• The general rule for a "greater than" situation is

P (greater than a value) = 1 - P(less than or equal to the value)

Example : Using Minitab we can find that the probability = 0.1587 that a speed is less than or equal to 60 mph. Thus the probability a speed is greater than 60 mph = 1 - 0.1587 = 0.8413.

The relevant Minitab output and a figure showing the cumulative probability for 60 mph follows:

"In between" Probabilities

Suppose we want to know the probability a normal random variable is within a specified interval. For instance, suppose we want to know the probability a randomly selected speed is between 60 and 73 mph. The simplest approach is to subtract the cumulative probability for 60 mph from the cumulative probability for 73. The answer is

Probability speed is between 60 and 73 = 0.9452 − 0.1587 = 0.7865.

This can be written as P(60 < X < 73) = 0.7865, where X is speed.

• The general rule for an "in between" probability is

P( between a and b ) = cumulative probability for value b − cumulative probability for value a

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Source: https://online.stat.psu.edu/stat800/book/export/html/659

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