... Borel real line
↑, ↓ increasing (non-decreasing) and decreasing (non-increasing), respectively
P, (S, A, P) probability measure (function) and probability space, respectively
I
A
indicator of ... found in
most books of linear algebra. For example, see Birkhoff and MacLane, A
Survey of Modern Algebra, 3d ed., MacMillan, 1965; S. Lang, Linear Algebra,
Addison-Wesley, 1968; D. C. Murdoch,...
... fertilizers.
Comparison of a certain brand of gasoline with and without an additive by
using it in several cars.
Comparison of different brands of gasoline by using them in several cars.
Comparison of the wearing ... distributed, under H. We may
summarize all relevant information in a table (Table 1) which is known as an
Analysis of Variance Table.
EXAMPLE 3
For a numerical exampl...
... Determine the minimum sample size n required by using the Binomial
tables (if possible) and also by using the CLT.
13.3 .10 The number X of fatal traffic accidents in a certain city during a year
may ... Testing Hypotheses
13.3.4 The rainfall X at a certain station during a year may be assumed to be
a normally distributed r.v. with s.d.
σ
= 3 inches and unknown mean
μ
. For the...
... permutation of the integers 1 through n. From
the definition of a determinant and the fact that each row and column contains
exactly one 1 and the rest all 0, it follows that the n! Jacobians are ... of a normally distributed r.v. with mean a
μ
+ b and variance
a
2
σ
2
. Thus, if X is N(
μ
,
σ
2
), then aX + b is N (a
μ
+ b, a
2
σ
2
).
Now it may happen that the transformation h sat...
... black
at least 2 cards in are red
exactly 1 card in is an ace
the first card in is a diamond,
the second is a heart and the third is a club
card in is a diamond, 1 is a heart and 1 is a club
2.4.30 ... is a generic model for situations often occurring in practice. For
instance, the urn and the balls may be replaced by a box containing certain
items manufactured by...
... repeated at will, in many other cases this is, indeed, the case. In
the latter case, as a rule, it is advantageous not to fix the sample size in
advance, but to keep sampling and terminate the ... (as well as in the interval estimation problems
to be dealt with in Chapter 15), sampling according to a stopping time is, in
general, profitable, the mathematical machinery involved...
...
()
∈
[]
=
()
∈
[]
By assuming the existence of an MLE, we have that the maximum at the
left-hand side above is attained at an MLE
ˆ
θθ
θθ
θ. Then, clearly, the right-hand
side attains its maximum at
ˆ
θθ
θθ
θ*, ... The Case Where Complete Sufficient Statistics Are Not Available or
May Not Exist: Cramér–Rao Inequality
When complete, sufficient statistics are available, the problem of findi...
... differentiation as follows:
178 7 Stochastic Independence with Some Applications
To start with, consider the probability space (S, A, P) and recall that k
events A
1
, , A
k
are said to be independent ... a function defined on the entire real line
and taking values in the complex plane. Those readers who are not well versed
in matters related to complex-valued functions may feel un...
... improper integral in which the limits
are taken symmetrically. As an example, for
σ
= 1,
μ
= 0, we have, in terms of
the principal value integral,
I
xdx
x
x
AA
A
A
A
A
A
A
A
* lim lim log
lim ... over x, as x varies
in G, of such intervals is equal to G. The same is true if we consider only those
intervals corresponding to all rationals x in G. These intervals are countably...
... .
n
nj
j
AA
→∞
=
∞
=
1
U
We recall that
AAAA AAA
AAA AA
j
j
ccc
=
∞
=+ ∩
()
+∩∩
()
+⋅⋅⋅
=+ −
()
+−
()
+⋅⋅⋅
1
11 2 1 23
121 32
U
,
by the assumption that A
n
↑. Hence
P A P A PA PA A
PA A PA A
PA PA A ... n
n
nn
n
nn
23
2
10
21
⎛
⎝
⎜
⎞
⎠
⎟
+
⎛
⎝
⎜
⎞
⎠
⎟
+⋅⋅⋅+
⎛
⎝
⎜
⎞
⎠
⎟
=−
⎛
⎝
⎜
⎞
⎠
⎟
−
⎛
⎝
⎜
⎞
⎠
⎟
=−−
relationships characterizing the independence of A
j
, j = 1, , n and the...