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27/11/2015 10:50:05 AM
Giảng dạy - Học tập
FURTHER APPLICATIONS OF INTEGRATION - PROBABILITY
Calculus plays a role in the analysis of random behavior. Suppose we consider the cholesterol
level of a person chosen at random from a certain age group, or the height of an adult female
chosen at random, or the lifetime of a randomly chosen battery of a certain type.
Such quantities are called continuous random variables because their values actually
range over an interval of real numbers, although they might be measured or recorded only
to the nearest integer. We might want to know the probability that a blood cholesterol level
is greater than 250, or the probability that the height of an adult female is between 60 and
70 inches, or the probability that the battery we are buying lasts between 100 and 200
hours. If X represents the lifetime of that type of battery, we denote this last probability as
follows:
According to the frequency interpretation of probability, this number is the long-run proportion
of all batteries of the specified type whose lifetimes are between 100 and 200
hours. Since it represents a proportion, the probability naturally falls between 0 and 1.
Every continuous random variable X has a probability density function . This means
that the probability that X lies between a and b is found by integrating from a to b:
For example, Figure 1 shows the graph of  a model for the probability density function f
for a random variable X defined to be the height in inches of an adult female in the
United States (according to data from the National Health Survey). The probability that the
height of a woman chosen at random from this population is between 60 and 70 inches is
equal to the area under the graph of f  from 60 to 70.
In general, the probability density function of a random variable X satisfies the condition
 f(x) greater than or equal to 0 for all x. Because probabilities are measured on a scale from 0 to 1, it follows
that