Page 67 - Probability and Statistical Inference

44 1. Notions of Probability

numbers a and b, we then have

This conveys the following message: Given that a component has lasted up
until the time a, the conditional probability of its surviving beyond the time
a + b is same as P(X > b}, regardless of the magnitude of a. In other words,
the life of the component ignores the aging process regardless of its own age.
This interesting feature of the exponential distribution is referred to as its
memoryless property. The recently edited volume of Balakrishnan and Basu
(1995) gives a synthesis of the gamma, exponential, and other distributions.
The Chi-square Distribution: We say that a positive continuous random
variable X has the Chi-square distribution with ν degrees of freedom denoted
by , with ν = 1, 2, 3, ..., if X has the Gamma(1/2ν, 2) distribution. Here,
the parameter ν is referred to as the degree of freedom. By varying the values
of ν, one can generate interesting shapes for the associated pdf.

Figure 1.7.6. PDF’s: (a) x 2 5 Thin; x 2 10 Thick (b) x 2 25 Thin; x 2 35 Thick

A Chi-square random variable is derived from the Gamma family and so it
should not be surprising to learn that Chi-square distributions are skewed to
the right too. In the Figure 1.7.6, we have plotted the pdf’s corresponding
of the random variable when ν = 5, 10, 25, 30. From these figures, it
should be clear that as the degree of freedom ν increases, the pdf tends to
move more toward the rhs. From the Figure 1.7.6 (b) it appears that the
shape of the pdf resembles more like that of a symmetric distribution when