Areas under continuous probability density functions

Let X be a continuous random variable with probability density function and cumulative distribution function . Then:

1. The total area under the curve is 1.

2. The area under the curve to the left of is ,

where is any value that the random variable X can take.

The area under the probability distribution curve of a continuous random variable between any two points is between 0 and 1, as shown in Figure 4.3.

The total area under the probability distribution curve of a continuous random variable is always 1.0 or 100% as shown in Figure 4.4.

Remark:

The probability that a continuous random variable x assumes a single value is always zero.

This is because the area of a line, which represents a single point,

is zero. (Fig.4.5)

In general, if a and b are two of the values that X can assume, then,

and .

When determining the probability of an interval a to b, we need not be concerned if either or both end points are included in the interval. Since the probabilities of and are both equal to 0,

.

Дата добавления: 2015-08-05