Continuous Probability Distributions Study Guide
Introduction to Continuous Probability Distributions
As was the case with discrete distributions, some continuous random variables are of particular interest. In this lesson, we will discuss two of these: the uniform distribution and the normal distribution. The normal distribution is particularly important because many of the methods used in statistics are based on this distribution. The reasons for this will become clearer as we work through the rest of the lessons.
In the first lesson, we learned that a continuous random variable has a set of possible values that is an interval on the number line. It is not possible to assign a probability to each point in the interval and still satisfy the conditions of probability set forth in Lesson 10 for discrete random variables. Instead, the probability distribution of a continuous random variable X is specified by a mathematical function f(x) called the probability density function or just density function. The graph of a density function is a smooth curve.A probability density function (pdf) must satisfy two conditions: (1) f(x) ≥ 0 for all real values of x and (2) the total area under the density curve is equal to 1. The graphs of three density functions are shown in Figure 11.1.
The probability that X lies in any particular interval is shown by the area under the density curve and above the interval. The following three events are frequently encountered: (1) X < a, the event that the random variable X assumes a value less than a; (2) a < X < b, the event that the random variable X assumes a value between a and b; and (3) X > b, the event that the random variable X is greater than b. We say that we are interested in the lower tail probability for (1) and the upper tail probability when using (3). The areas associated with each of these are shown in Figure 11.2.
Notice that the probability that a < X < b may be computed using tail probabilities:
P(a < X < b) = P(X < b) – P(X < a).
If the random variable X is equally likely to assume any value in an interval (a, b), then X is a uniform random variable. The pdf is flat and is above the x-axis between a and b, and it is 0 outside of the interval. The height of the curve must be such that the area under the density and above the x-axis is 1. Because this region is a rectangle, the area is the height times the width of the interval, which is b – a. Thus, the height must be ; that is, the pdf of a uniform random variable has the form
= 0, otherwise.
A graph of the pdf is shown in Figure 11.3.
A group of volcanologists (people who study volcanoes) has been monitoring a volcano's seismicity, or the frequency and distribution of underlying earthquakes. Based on these readings, they believe that the volcano will erupt within the next 24 hours, but the eruption is equally likely to occur any time within that period. What is the probability that it will erupt within the next eight hours?
Define X = the time until the eruption of the volcano. X has positive probability over the interval (0,24) because the volcano will erupt during that time interval. Because the length of the interval is 24 – 0 = 24, the height of the density curve must be for the area under the density and above the x-axis to be one. That is, the pdf is
= 0, otherwise.
The probability that the volcano will erupt within the next eight hours is equal to the area under the curve and above the interval (0,8) as shown in Figure 11.4. This area is .
In the previous example, notice that the area is the same whether we have P(0 < X < 8) or P(0 ≤ X < 8) or P(0 < X ≤ 8) or P(0 ≤ X ≤ 8). Unlike discrete random variables, whether the inequality is strict or not, the probability is the same for the continuous random variables. This also correctly implies that, for continuous random variables, the probability that the random variable equals a specific value is 0.
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