Calculation Procedure:- Write the equation of cumulative probability
Refer to the calculation procedure on the normal distribution for definitions pertaining to
a continuous random variable. A variable X is said to have a negative-exponential (or sim-
ply exponential) probability distribution if its probability density function is of this form:
f(X) = O if X< O and/^Y) = ae'^if X > O Eq. a, where a = positive constant and e = base
of natural logarithms. Figure 22 shows the probability diagram. The arithmetic mean of X
is jji = I/a, Eq. b.
Let X = life span of device, months, and let K denote any positive number. Integrate
Eq. a between the limits of O and K, giving P(X < K)=I- e~aK, Eq. c. Then P(X >K) =
e-°K, Eq. d. - Compute the required probability
Compute a by Eq. b, giving a = I/JJL = 1/2 = 0.5. Set K = 3 months. By Eq. d, P(X> 3) =
^r^1 -^5 = !/*?^1 -^5 = 0.2231.
Statistical InferenceConsider that there exists a set of objects, which is called the population, or universe.
Also consider that interest centers on some property of these objects, such as length, mo-
lecular weight, etc., and that this property assumes many values. Thus, associated with the
population is a set of numbers. This set of numbers has various characteristics, such as
arithmetic mean and standard deviation. A characteristic of this set of numbers is called a
parameter. For example, assume that the population consists of five spheres and that they
have the following diameters: 10, 13, 14, 19, and 21 cm. The diameters have an arith-
metic mean of 15.4 cm and standard deviation of 4.03 cm, and these values are parame-
ters of the given population.
Now consider that a subset of these objects is drawn. This subset is called a sample,
and a characteristic of the sample is called a statistic. Thus, using the previous illustration,
assume that the sample consists of the spheres having diameters of 14, 19, and 21 cm.
These diameters have an arithmetic mean of 18 cm and standard deviation of 2.94 cm,
and these values are statistics of the sample drawn. The number of objects in the sample is
the sample size.
FIGURE 22. Negative-exponential probability distribution.