Page 87 - Probability and Statistical Inference

64 1. Notions of Probability

1.7.22 Consider an arbitrary random variable Y which may be discrete or
continuous. Let A be an arbitrary event (Borel set) defined through the ran-
dom variable Y. For example, the event A may stand for the set where Y ≥ 2 or
|Y| > 4 ∪ |Y| ≤ 1/2 or one of the many other possibilities. Define a new random
variable X = I(A), the indicator variable of the set A, that is:

Argue that X is a Bernoulli variable, defined in (1.7.1), with p = P(A).
1.7.23 Consider the negative exponential pdf

+
from (1.7.36) where β ∈ ℜ , γ ∈ ℜ. Plot the pdf for several values of β and
γ. Answer the following questions by analyzing these plots.
(i) If β is held fixed, will P {X > 3} be larger than
γ=1
P {X > 3}?
γ=2
(ii) If γ is held fixed, will P {X > 4} be larger than
β=2
P {X > 4}?
β=3