The Probability Distribution Of A Continuous Random Variable

The Probability Distribution of a Continuous Random Variable

For a discrete random variable X the probability that X 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 X denotes the length of time a commuter just arriving at a bus stop has to wait for the next bus. If buses run every 30 minutes without fail, then the set of possible values of X is the interval denoted [0,30], the set of all decimal numbers between 0 and 30. But although the number 7.211916 is a possible value of X, there is little or no meaning to the concept of the probability that the commuter will wait precisely 7.211916 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 7.211916 minutes. More meaningful questions are those of the form: What is the probability that the commuter's waiting time is less than 10 minutes, or is between 5 and 10 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.

Definition

The probability distribution of a continuous random variable X is an assignment of probabilities to intervals of decimal numbers using a function f(x), called a density functionThe function f(x) such that probabilities of a continuous random variable X are areas of regions under the graph of y=f(x)., in the following way: the probability that X assumes a value in the interval [a,b] is equal to the area of the region that is bounded above by the graph of the equation y=f(x), bounded below by the x-axis, and bounded on the left and right by the vertical lines through a and b, 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 1, and the proportion of the population with measurements between two numbers a and b is the area under the curve and between a and b, as shown in Figure 2.6 "A Very Fine Relative Frequency Histogram" in Chapter 2 "Descriptive Statistics". If we think of X as a measurement to infinite precision arising from the selection of any one member of the population at random, then P(a<X<b) is simply the proportion of the population with measurements between a and b, the curve in the relative frequency histogram is the density function for X, and we arrive at the definition just above.

Every density function f(x) must satisfy the following two conditions:

1. For all numbers x, f(x)≥0, so that the graph of y=f(x) never drops below the x-axis.
2. The area of the region under the graph of y=f(x) and above the x-axis is 1.

Because the area of a line segment is 0, the definition of the probability distribution of a continuous random variable implies that for any particular decimal number, say a, the probability that X assumes the exact value a is 0. This property implies that whether or not the endpoints of an interval are included makes no difference concerning the probability of the interval.

For any continuous random variable X:

P(a≤X≤b)=P(a<X≤b)=P(a≤X<b)=P(a<X<b)

Example 1

A random variable X has the uniform distribution on the interval [0,1]: the density function is f(x)=1 if x is between 0 and 1 and f(x)=0 for all other values of x, as shown in Figure 5.2 "Uniform Distribution on ".

Figure 5.2 Uniform Distribution on [0,1]

1. Find P(X > 0.75), the probability that X assumes a value greater than 0.75.
2. Find P(X ≤ 0.2), the probability that X assumes a value less than or equal to 0.2.
3. Find P(0.4 < X < 0.7), the probability that X assumes a value between 0.4 and 0.7.

Solution:

1. P(X > 0.75) is the area of the rectangle of height 1 and base length 1−0.75=0.25, hence is base×height=(0.25)·(1)=0.25. See Figure 5.3 "Probabilities from the Uniform Distribution on "(a).
2. P(X ≤ 0.2) is the area of the rectangle of height 1 and base length 0.2−0=0.2, hence is base×height=(0.2)·(1)=0.2. See Figure 5.3 "Probabilities from the Uniform Distribution on "(b).
3. P(0.4 < X < 0.7) is the area of the rectangle of height 1 and length 0.7−0.4=0.3, hence is base×height=(0.3)·(1)=0.3. See Figure 5.3 "Probabilities from the Uniform Distribution on "(c).

Figure 5.3 Probabilities from the Uniform Distribution on [0,1]

Example 2

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.

Solution:

The graph of the density function is a horizontal line above the interval from 0 to 30 and is the x-axis everywhere else. Since the total area under the curve must be 1, the height of the horizontal line is 1/30. See Figure 5.4 "Probability of Waiting At Most 10 Minutes for a Bus". The probability sought is P(0≤X≤10). By definition, this probability is the area of the rectangular region bounded above by the horizontal line f(x)=1∕30, bounded below by the x-axis, bounded on the left by the vertical line at 0 (the y-axis), and bounded on the right by the vertical line at 10. This is the shaded region in Figure 5.4 "Probability of Waiting At Most 10 Minutes for a Bus". Its area is the base of the rectangle times its height, 10·(1∕30)=1∕3. Thus P(0≤X≤10)=1∕3.

Figure 5.4 Probability of Waiting At Most 10 Minutes for a Bus