Page 55 - Probability and Statistical Inference

32 1. Notions of Probability

x = (x , x ) ∈ (a , b ) × (a , b ) ⊆ ℜ . The process of finding where this
2
2
1
2
1
1
2
function f(x) attains its maximum or minimum requires knowledge of matri-
ces and vectors. We briefly review some notions involving matrices and vec-
tors in the Section 4.8. Hence, we defer to state this particular result from
calculus in the Section 4.8. One should refer to (4.8.11)-(4.8.12) regarding this.
Integration by Parts: Consider two real valued functions f(x), g(x) where
x ∈ (a, b), an open subinterval of ℜ. Let us denote d/dx f(x) by f (x) and the
indefinite integral ∫ g(x)dx by h(x). Then,
assuming that all the integrals and f’ (x) are finite.
L’Hôpital’s Rule: Suppose that f(x) and g(x) are two differentiable real
valued functions of x ∈ ℜ. Let us assume that and
where a is a fixed real number, –∞ or +∞. Then,

where f’ (x) = df(x)/dx, g’ (x) = dg(x)/dx.
Triangular Inequality: For any two real numbers a and b, the following
holds:

From the triangular inequality it also follows that

One may use (1.6.30) and mathematical induction to immediately write:

where a , a , ..., a are real numbers and k = 2.
1 2 k
1.7 Some Standard Probability Distributions

In this section we list a number of useful distributions. Some of these distri-
butions will appear repeatedly throughout this book.
As a convention, we often write down the pmf or the pdf f(x)
only for those x ∈ χ where f(x) is positive.