Confidence Builder: Confidence Intervals/Sample Sizes
Please answer all questions fully and show all work/calculations. Do not write
answers/calculations on the question paper; no credit will be given for
171 Final Solutions, Fall 20161
1. Question 1
(a) Let cfw_N (s)s0 be a Poisson Process with parameter = 1. Then
N (4) N (3), N (3) N (2), N (2) N (1), N (1)
are all independent random variables.
TRUE. This is the independent increment property,
171 Midterm 2 Solutions, Fall 20161
1. Question 1
(a) Every Markov chain has at most one stationary distribution.
FALSE. Consider the Markov chain with transition matrix P =
. Then = (1, 0)
and = (0, 1) are both distinct stationary dist
171 Midterm 1 Solutions, Fall 20161
1. Question 1
(a) Let be a universe. Let A1 , A2 , . . . . Then
Ai = cfw_x : positive integers j, x Aj .
FALSE. If A1 = and A2 = , then
i=1 Ai = , but = cfw_x : positive integers j, x
(b) For a
Limits Using the Definition
Properties of Absolute Values & Inequalities
1) If x a < , then < x a <
a < x < a + .
MF M F < K
2) If M < and F < K then =
3) Triangle Inequality: | A + B | | A | + | B |
4) | A | | B | | A B | | A | + | B |
Journal #4 Women in Modern America
Society Proceedings by Dr. Cushing
1. What womens cases are Cushing and the medical professionals discussing?
It is a case of Melancholia, masturbation, cured by removal of both ovaries.
Patient is a 33 years o
mg i'heorem 1.6.1: If two lines are perpendicular, then they meet to form right angles
33 ('TheOrem 1.7.1: if two lines meet to form a right angle, then these lines are perpendicular
96/ Theorem 1.7.2: If two angles are complementary to the same angle
f Corollary 3.3.6: An equiangular triangle is also equilateral.
/ The perimeter of a triangle is the sum of the lengths of its sides,
Ch awe l E) . E) Inequalities in a Triangle
Lemma 3.5.1: If B is between A and C on line AC, then AC > AB and
Theorem 5.4.1: The altitude drawn to the hypotenuse of a right triangle separates the right triangle into two right triangles that are
similar to each other and to the original right triangle.
Theorem5.4.2: The length of the altitude to the hypotenuse of
.,"l ~ 1.- gm - mm-W nun-ax.
t . 4._ _ -. u-._ .-.,. -rt
secant a = .
h otenuse' ad acent '
., cotangent a = -,
ad acent o lostte
cosine a =
tangent a =
Applications with Acute Triangle
Theorem 1 1.4.1: The a
. new 1-H
in this context, ldentity is the reason we cite when verifying that afne segment or an angle is congruent to itself; also known as the
Reexive Property of Congruence.
Postulate 14: if two angles and the included side of one triangle are co
1. An isosceles triangle is a triangle that has two congruent sides
2. A line segment is the part of a line that consists of two points, known as endpoints, and all points between them.
3. A postulate is a statement that is assumed to be true.
(LAOPW Y W" 1 Ratios, Rates, and Proportions
A rate is a quotient that compares two quantities that are incommensurable.
A proportion is a statement that two ratios or two rates are e ual. - - =
Property 1(means extremes property): in a proportion, the pg
Denition: IlP, then Q known as a conditional statement (or implication)-
Denition: If P, then Q P is the hypothesis and Q is the conclusion.
Hypothesis: Two sides of a triangle are equal in length
Conclusion: Two angles of the triangle are equal in measur