Page 73 - Probability and Statistical Inference

50 1. Notions of Probability

The Weibull Distribution: We say that a positive continuous random
variable X has the Weibull distribution if and only if its pdf is given by

where α(> 0) and β(> 0) are referred to as parameters. This pdf is also
skewed to the right. By varying the values of α and β, one can generate
interesting shapes for the associated pdf. The Weibull pdf from (1.7.37) has
been plotted in the Figure 1.7.12 for (α, β) = (3, .5), (3, 1). These figures are
skewed to the right too.
The Rayleigh Distribution We say that a positive continuous random
variable X has the Rayleigh distribution if and only if its pdf is given by

where θ(> 0) is referred to as a parameter. In reliability studies and related
areas, this distribution is used frequently.

Figure 1.7.13. Raleigh Densities: (a) θ = 1 (b) θ = 4

The pdf from (1.7.38) has been plotted in the Figure 1.7.13 for θ = 1, 4. This
pdf is again skewed to the right. The tail in the rhs became heavier as θ in-
creased from 1 to 4.

1.8 Exercises and Complements

1.1.1 (Example 1.1.1 Continued) In the set up of the Example 1.1.1, find
the probability of observing no heads or two heads.
1.1.2 (Example 1.1.2 Continued) In the set up of the Example 1.1.2, find
the probability of observing a difference of one between the scores which