Math 140, Dr. McLoughlin, Axioms of Probability Hand-out , Page 1 of 1
HAND-OUT 2
PROBABILITY
M. P. M. M. MCLOUGHLIN
Recall from Set Theory that elements, a, are members of sets, A, and a set, A, is a subset
of some well defined universe, U.
The universe
MATH 140 SUMMER 2010 WORKSHEET 1 NAME:_
(please print legibly)
Let U = S =
(the sample space is the integers).
Let the data be described as X1, X2, X3, X4 (as in problem 1), as X1, X2, X3, X4, X5 (as in problem 2) etc.
0. Let the data set be D0 = cfw_1,
MATH 140
Let D
WORKSHEET
NAME:_
= cfw_x1, x2, x3, x4, x5, . . . , xn where n is a natural number (n ).
Consider the formulae:
n
X
X
k
k 1
n
gD
n
n
X
k 1
k
hD
n
1
X
k 1
k
n
Find each of the following:
1. X , g D , and h D where D
= cfw_1, 2, 4
2.
MATH 140 SUMMER 2012 WORKSHEET
NAME:_
(please print legibly)
For quantitative questions 1 3 U = (the universe is the real numbers) Let S = (the sample space is
the integers). So, for any function, f, the domain will be a subset of and the codomain is
(s
MATH 140 SUMMER 2012 WORKSHEET 1
NAME:_
(please print legibly)
Please determine the following:
A.
B.
Which of the following experiments has a well defined sample space (that means satisfies the axioms of probability)?
Which of the following experiments i
MATH 140 SUMMER 2009 WORKSHEET II NAME:_
(please print legibly)
Let U = S =
(the sample space is the integers).
Let the data be described as X1, X2, X3, X4 (as in problem 1), as X1, X2, X3, X4, X5 (as in problem 2) etc.
1. Let the data set be D1 = cfw_1,
MATH 140 WORKSHEET VIII FALL 2011
1. Consider we have a researcher who is interested in studying the average weight of Pennsylvania cats as opposed to
the general cat population. He sets = .05. He finds that the National Institute of Standards and Technol
MATH 140 WORKSHEET VI FALL 2011 REVISED 2013
1A. Suppose Farmer McLoughlin is concerned about the cost to produce a bushel of pole beans
crop. Suppose (a huge assumption) that the true population is normally distributed with a mean
for the cost to produce
MATH 140 WORKSHEET VII FALL 2011, page 1 of 2
MATH 140 WORKSHEET VII FALL 2011
1. Consider we have a researcher who is interested in studying the average weight of Pennsylvania cats as opposed to
the general cat population. He sets = .05. He finds that th
MATH 140 SUMMER 2010 WORKSHEET II
NAME:_
(please print legibly)
Let U = S =
Z (the sample space is the integers).
Let the data be described as X1, X2, X3, X4 (as in problem 1), as X1, X2, X3, X4, X5 (as in problem 2) etc.
1. Let the data set be D1 = cfw_
MATH 140
WORKSHEET ZERO
NAME:_
Find each of the following:
1. Consider the function f (k) 2k 3 where f : find the first 6 terms of f.
n
2. Consider the formula
f (k)
k 1
n
. Let n = 5 and f (k) 2k 3 . Find
n
3. Consider the formula
p f (k) . Let n = 3
MATH 140 WORKSHEET IV FALL 2011 NAME:_
(PLEASE PRINT LEGIBLY)
1. Suppose the random variable X is a binomial experiment with 5 repetitions and the probability of success is 0.5
A. Find the probability of exactly 2 successes.
B. Find the probability of exa
MATH 140 WORKSHEET IV SUMMER 2011
NAME:_
(PLEASE PRINT LEGIBLY)
1. Suppose we have U =
1. Suppose we have S
= 7 (the sample space consists of real numbers in the set 7).
Pick a number:
A. Find the probability of picking an even number.
B. Find the probab
MATH 140 FALL 2011 WORKSHEET I
NAME:_
(please print legibly)
Let U = S =
(the sample space is the integers).
Let the data be described as as X1, X2, X3, X4, X5 (as in problem 1) etc.
1. Let the data set be D1 = cfw_1, 2, 2, 3, 4
Find:
A. the mode of the
MATH 140 SUMMER 2009 WORKSHEET I
NAME:_
(please print legibly)
Let U = S =
(the sample space is the integers).
Let the data be described as X1, X2, X3, X4 (as in problem 1), as X1, X2, X3, X4, X5 (as in problem 2) etc.
1. Let the data set be D1 = cfw_1,
Dr. McLoughlins Math 140 Correlation, page 1 of 4
MATH 140
DR. MCLOUGHLINS CLASS
STATISTICAL FORMULAE FOR CORRELATION
HANDOUT V PART B
Recall we are attempting to estimate parameters in a population (e.g.: a population mean ,
a population variance , a pop
Math 140 - Dr. McLoughlins Class Bernoulli / Binomial, page 1 of 3
MATH 140
DR. MCLOUGHLINS CLASS BERNOULLI AND
BINOMIAL RANDOM VARIABLES
HANDOUT III
Note: The basic building block of probability is set theory:
Suppose we have a well defined sample space
Math 140 - McLoughlins Class Measurement Scales, page 1 of 4
MATH 140
DR. MCLOUGHLINS CLASS
MEASUREMENT SCALES
HANDOUT I
A whole point to studying probability and statistics is the analysis of data. So, when
one is studying variables it is important to un
Dr. McLoughlin Math 140 Regression Formulae Draft 1, page 1 of 2
MATH 140
DR. MCLOUGHLINS CLASS
REGRESSION FORMULAE
HANDOUT VI DRAFT 1
Let U = S be a well defined universe (the sample space) for our work S will always be able to
be a subset of (the reals,
Math 140 - McLoughlins Class Measures of Central Tendency, page 1 of 5
MATH 140
DR. MCLOUGHLINS CLASS
MEASURES OF CENTRAL TENDENCY OR LOCATION
HANDOUT II
Now, we are studying probability and statistics. We are considering random
variables as defined in ha
Math 140 Sampling Dr. McLoughlins Class Handout 5, page 1 of 2
MATH 140 DR. MCLOUGHLINS CLASS
SOME SAMPLING DISTRIBUTION RESULTS
HANDOUT V
Let n, p, , , , , , , , c be constants.
Theorem 1 (DeMoivre - Laplace) : Let X ~ Bin (x, n, p). Let Y =
X np
.
np(1
Dr. McLoughlin Math 140 Statistical Formulae for Test 1, page 1 of 1
MATH 140
DR. MCLOUGHLINS CLASS
STATISTICAL FORMULAE FOR THE TEST HANDOUT
Let U = S be a well defined universe (the sample space) for our work S will always be able to
be a subset of (the
Dr. McLoughlin Math 140 Statistical Formulae, page 1 of 2
MATH 140
DR. MCLOUGHLINS CLASS
STATISTICAL FORMULAE
HANDOUT IV
Let U = S be a well defined universe (the sample space) for our work S will always be able to
be a subset of (the reals, of course).
L
Dr. McLoughlins Math 140 T Formulae, page 1 of 4
MATH 140
DR. MCLOUGHLINS CLASS
STATISTICAL FORMULAE FOR T-TEST
HANDOUT V (REVISED AND EXPANDED)
We are called upon many times to determine if the mean performance of two groups are significantly
different.
Dr. McLoughlins Math 140 Formulae Test 2, page 1 of 2
MATH 140
DR. MCLOUGHLINS CLASS
STATISTICAL FORMULAE FOR TEST 2
Let D = cfw_X1, X2, X3, . . . , Xn be a finite data set from a population of interest and
C = cfw_Y1, Y2, Y3, . . . , Yn be a finite data
Dr. McLoughlins Math 140 T Formulae, page 1 of 1
MATH 140
DR. MCLOUGHLINS CLASS
STATISTICAL FORMULAE FOR TEST 3
n
X
T
(X k X)2
S2
X
k 1
n
k 1
n 1
SX
n
X
XX
SX
Tn 1
SX
n
n
Xk
X
n
SX
Zi
(X
k 1
k
X)2
(n 1)
Xi X
X
Z
X X
X
X X
SX
Independent t-test P
Math 224 Dr. McLoughlins Inductive Reasoning Exercise, page 1 of 3
FOUNDATIONS OF MATHEMATICS
DR. MCLOUGHLINS CLASS
EXERCISE SET 0
1. The Circle Problem
Let C be a circle with radius one centred at (0,0).
Step 1. Let A1 and A2 be points on the circle. Cal
Math 140 Worksheet 5 - Dr. McLoughlins Class, page 1 of
MATH 140 FALL 2011
WORKSHEET V
Show all work. If an answer DNE, explain why it does not exist.
Note the following abbreviations:
On this worksheet het X ~ Nor(x, , )
1. Suppose we have X ~ Nor(x, 4,