Introduction to Probability and
Statistics
Two-Sample Tests
Z test for Difference Between
Two Means When is known
For given independent samples, we want to test hypothesis for the
difference between two population means, 1 2 , using the following
Test sta
Introduction to Probability and
Statistics
Basic Probability
Learning Objectives
n
n
n
n
Basic probability concepts
Various counting rules
Conditional probability
To use Bayes Theorem to revise probabilities
Basic Probability Concepts
n
n
n
Probability th
Introduction to Probability and Statistics in the Life Sciences
Answers to certain Exercises from Chapter 2
1. (a) cfw_-1,1,3
(b) cfw_-15,-13,-11,-9,-7,-5,-3,-2,-1,0,1,2,3,5,7,9,11,13,15
(c) cfw_ x U | x is even
(d) cfw_-14,-12,-10,-8,-6,-4,4,6,8,10,12,1
Statistics 101 &
Exploratory Data Analysis (EDA)
A definition of probability
Consider a set S with subsets A, B, .
Kolmogorov
axioms (1933)
From these axioms we can derive further properties, e.g.
Conditional probability, independence
Also define conditio
FROM A
x
PDF: fX(x; )=
(1- )
where 0< <1, 0
:
1- x
Note that each Xi in the SUM must be
th
either a 1 or a 0 accordingly as the i
object selected is, or is not, a
, i=1,2,.,n.
Thus, in this case,
T actually represents the
TOTAL NUMBER of
in the HYPOTHETI
PARAMETER:
UNBIASED
ESTIMATOR: G
TRUE ARRIVAL TIME: 9:00 am
Observed Mean Arrival Time
Moe: 9:15 am Curly: 9:00 am Larry: 8:50 am
Positively Biased
Unbiased
Negatively Biased
Most Efficient
Moe: 10 min
Curly: 1 min Larry: 5 min
Variation in Arrival Time
Suppose we have a
POPULATION
indexed by a
RANDOM VARIABLE, X
[
or
]
having TRUE
Watson, its high time we solved the
My feelings entirely, Holmes!
DENSITY CURVE (p.d.f.)
2 -1/2
2
2
fX(x; , )=(2 ) expcfw_-(x- ) /2
An ESTIMATOR of the form
Also termed a
!
MA241 Lab Report 1 - Sets and Probability
Name:
Student Number:
Lab
Winter 2016
1. [6 marks] The obstetrics ward at Doublin General Hospital is world-renowned for its care and treatment of
multiple-birth pregnancies, especially twins. Last year, the staff
MA241 Lab Report 2 - Discrete and Continuous Random Variables
Name:
Student Number:
Lab
Winter 2016
1. [2 marks] Classify the following random variables as discrete or continuous by circling the correct answer.
(a) H: the height of 5-year-olds in Canada.
MA241 Lab 3 Notes
Text Reference: 3.2.2, 3.4.1, 5.2.2, 5.2.2.1, 5.3, 5.3.1
Continuous Random Variables (3.2.2)
See Lab 2 Notes.
Chebyshevs Theorem (3.4.1)
Suppose the random variable X has finite mean and variance 2 . Then, for any k > 0, the following h
MA241 Lab 1 Notes
Text Reference: 2.2 - 2.5, 2.7
Sets (2.7)
A set S is a collection of objects or elements.
A is a subset of S, written A S, if S contains all members of A.
The empty set or null set is the set consisting of no elements and is denoted b
Introduction to Probability and
Statistics
Simple Linear Regression
Scatterplots and Correlation
Scatterplot
Example: Make a scatterplot of the relationship between body
weight and pack weight for a group of hikers.
Body weight (lb)
120
187
109
103
131
1
Introduction Probability and Statistics in the Life Sciences
Chapter 3: Answers to Selected Exercises
2. (a) k = 70/197
X=x
1
2
3
4
5
P (X = x) 70/197 35/197 210/985 140/591 350/1379
(b) To ve decimals: E(X) = 3.32995 and V ar(X) = 1.79468
(c) (i) 350/118
Introduction to Probability and
Statistics
Conditional Probability
Conditional Probabilities
n
A conditional probability is the probability of one
event, given that another event has occurred:
P(A and B)
P(A | B) =
P(B)
The conditional
probability of A gi
Introduction to Probability and
Statistics
Joint Probability Distributions
for Bivariate and Mathematical
Expectations
Definitions
Definition
Definitions
Example
qWhat is the marginal distribution of X?
qWhat is the marginal distribution of Y?
qWhat is th
Introduction to Probability and
Statistics
Special Discrete Probability
Distributions
Chap 4-1
Discrete Uniform Distribution
n
n
Let the discrete random variable X have non-zero
probability at each number x 1 < x 2 < < x n, n .
with P(X = x i ) = 1 / n .
Introduction to Probability and
Statistics
Some Continuous Probability
Distributions
Some Preliminaries
Moments and Their Use
n
Non-central moment
'
+
k
= x
k
f (x)dx
k
+
n
Central moment
n
Chebyshevs Inequality: for a positive k
= (x ) f (x)dx
k
P(| X
Introduction to Probability and
Statistics
Maximum Likelihood
Estimation and Sampling
Distributions
Point Estimate and Estimator
n
Definitions:
n
n
n
Let be a parameter to be estimated. A numerical
value, based on data collected in an experiment,
used to
Introduction to Probability and
Statistics
Graphical Techniques For Numerical
Data, Measure of Location, Measure of
Central Tendency
Visualizing Numerical Data:
The Histogram
A vertical bar chart of the data in a frequency distribution is
called a histogr
Introduction to Probability and
Statistics I
Data Variability Measure
Measures of Variation
Variation
Range
n
Variance
Standard
Deviation
Coefficient
of Variation
Measures of variation give
information on the spread
or variability or
dispersion of the dat
From Data Exploration to
Statistical Inference
Simple Random Sample (SRS)
The Idea of a Confidence Interval
4
Statistical Inference
After we have selected a sample, we know the responses of the individuals in the
sample. However, the reason for taking the
Introduction to Probability and
Statistics
Fundamentals of Hypothesis
Testing: One-Sample Tests
The Reasoning of Tests of Signicance
Suppose a basketball player claimed to be an 80% free-throw shooter. To test this
claim, we have him attempt 50 free-thro
Introduction to Probability and
Statistics
Chi-Square Tests
Contingency Tables
Contingency Tables
n
n
n
Useful in situations comparing multiple
population proportions
Used to classify sample observations according
to two or more characteristics
Also calle
Introduction to Probability and
Statistics
One-Way ANOVA
Introduction
The two sample t procedures compared the means of two
populations or the mean responses to two treatments in an
experiment.
In this chapter well compare any number of means using
Analys
MA241 Lab 2 Notes
Text Reference: 3.1, 3.2, 3.4, 3.5, 4.2.1, 5.2.1
Discrete Random Variables and Probability Distributions (3.1, 3.2.1)
A random variable is a variable (typically represented by X) that has a single numerical value, determined
by chance,