Probability Distribution for a Continuous Random Variable (CRV) . The probability distribution of
a continuous random variable has several properties.
• There is a smooth curve, called a density curve (defined by a density function ), that describes the
probability distribution of a CRV (sometimes called a probability distribution function). A density
curve is always on or above the horizontal axis (that is, it is always nonnegative) and has a total area
of 1 underneath the curve and above the axis.
• The probability of any individual value is 0. That is, if a is a point on the horizontal axis, P (X = a ) =
0.
• The probability of a given event is the probability that x will fall in some given interval on the
horizontal axis and equals the area under the curve and above the interval. That is, P (a < X < b )
equals the area under the graph of the curve and above the horizontal axis between X = a and X = b .
• The previous two bulleted items imply that P (a < X < b ) = P (a ≤ X ≤ b ).
In this course, there are several CRVs for which we know the probability density functions (a
probability distribution defined in terms of some density curve). The normal distribution (introduced in
Chapter 4 ) is one whose probability density function is the normal probability distribution . Remember
that the normal curve is “bell-shaped” and is symmetric about the mean, μ , of the population. The tails of
the curve extend to infinity, although there is very little area under the curve when we get more than, say,
three standard deviations away from the mean. (The 68-95-99.7 rule stated that about 99.7% of the terms
in a normal distribution are within three standard deviations of the mean. Thus, only about 0.3% lie
beyond three standard deviations of the mean.)
Areas between two values on the number line and under the normal probability distribution
correspond to probabilities. In Chapter 4 , we found the proportion of terms falling within certain
intervals. Because the total area under the curve is 1, in this chapter we will consider those proportions to
be probabilities.
Remember that we standardized the normal distribution by converting the data to z -scores
We learned in Chapter 4 that a standardized normal distribution has a mean of 0 and a standard