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BMEN

Clinical Trials & Regulations

1st half

Probability and Statistics

Semester 1, 2019

A/Prof. Leigh Johnston

Relevant Sections of Textbook BMEN Probability, Statistics and Stochastic Processes by Olofsson and Andersson List of sections covered in Probability & Statistics section Chapter 1 Basic Probability Theory 1.1 Introduction 1.2 Sample Spaces and Events 1.3 The Axioms of Probability 1.4 Finite Sample Spaces (not completely Combinatorics) 1.5 Conditional Probability and Independence 1.6 The Law of Total Probability and Bayes’ Formula Chapter 2 Random Variables 2.1 Introduction 2.2 Discrete Random Variables 2.3 Continuous Random Variables 2.4 Expected Value and Variance 2.5 Special Discrete Distributions (not 2.5.1, 2.5.3, 2.5.5 or 2.5.6) 2.6 The Exponential Distribution 2.7 The Normal Distribution 2.8.2 The Gamma Distribution Chapter 3 Joint Distributions 3.1 Introduction 3.2 The Joint Distribution Function 3.3 Discrete Random Vectors 3.4 Jointly Continuous Random Vectors 3.5 Conditional Distributions and Independence 3.7 Conditional Expectation (not beyond 3.7.1) 3.8 Covariance and Correlation 3.9 The Bivariate Normal Distribution Chapter 4 Limit Theorems 4.1 Introduction 4.2 The Law of Large Numbers 4.3 The Central Limit Theorem Chapter 6 Statistical Inference 6.1 Introduction 6.2 Point Estimators 6.3 Confidence Intervals (not 6.3.2 or 6.3.3) 6.5 Hypothesis Testing (not 6.5.1 or 6.5.2) 6.6.1 P-values 6.6.4 Multiple Hypothesis Testing Chapter 7 Linear Models 7.1 Introduction 7.2 Sampling Distributions 7.3 Single Sample Inference 7.4 Comparing Two Samples 7.5 Analysis of Variance (not 7.5.2) 7.6 Linear Regression 7.7 The General Linear Model

Why probability & statistics?

๏ (^) Probability: ๏ (^) How to design a new hearing aid that suppresses noise? ๏ (^) Medical Imaging: How to reduce the noise in the images? ๏ (^) Statistics: An example clinical trial ๏ (^) http://www.australiancancertrials.gov.au/search-clinical-trials/ search-results/clinical-trials-details.aspx? TrialID=365068&ds= ๏ (^) How to make sense of the results?

Experiments:

Procedures and Observations

๏ (^) The most basic notion of probability theory: ๏ (^) Performing an experiment ๏ (^) Experiment = a specified procedure and associated observation(s) ๏ Experiment may be repeatable, or may be one-off. ๏ If repeatable, the execution of the experiment is a trial. ๏ (^) Example: coin toss (repeatable) ๏ Procedure = toss a coin ๏ Observation = observing whether the coin lands heads or tails up ๏ Example: speculating on tech stocks in 1999 (non-repeatable!) ๏ Procedure = buy $XX,XXX worth of tech shares in 1999 ๏ Observation = read $ value of stock 3 years later…

More sample space examples

๏ (^) Example 1.3: Turn on a lightbulb and measure its lifetime, that is, the time until it fails. ๏ S =? ๏ (^) Example 1.4: Flip a coin twice and observe the sequence of heads and tails. ๏ S =? ๏ (^) Example 1.5: Throw a dart at random on a dartboard of radius r and observe its position. ๏ S =?

Events

๏ (^) A subset of the sample space is called an event. ๏ (^) An experiment yields only one outcome. ๏ (^) An event, A , is a collection of outcomes. ๏ (^) Two special events: ๏ (^) S itself is an event that always occurs ๏ (^) The empty set, ∅, never occurs.

A ✓ S

Set theory and Venn diagrams

๏ (^) Venn diagrams are a helpful way to illustrate sets and operations on them:

Set operations

๏ (^) Let A, B, and C be events. Then ๏ (^) Distributive Laws ๏ (^) De Morgan’s Laws

(A \ B) [ C = (A [ C) \ (B [ C)

(A [ B) \ C = (A \ C) [ (B \ C)

(A [ B)

c = A c \ B c (A \ B) c = A c [ B c

Example 1.

๏ (^) Mrs Boudreaux and Mrs Thibodeaux are chatting over their fence when the new neighbor walks by. He is a man in his sixties with shabby clothes and a distinct smell of cheap whiskey. Mrs B, who has seen him before, tells Mrs T that he is a former Louisiana state senator. Mrs T finds this very hard to believe. “Yes,” says Mrs B, “he is a former state senator who got into a scandal long ago, had to resign and started drinking.” “Oh,” says Mrs T, “that sounds more probable.” “No,” says Mrs B, “I think you mean less probable.” ๏ (^) Reasoning: ๏ A = {he is a former state senator} ๏ B = {he got into a scandal on ago, had to resign, and started drinking} ๏ (^) Write down Mrs B’s statements in terms of A and B, and prove her logic.

Example 1.

๏ (^) Consider the following statement: “I heard on the news that there is a 50% chance of rain on Saturday and a 50% chance of rain on Sunday. Then there must be a 100% chance of rain on the weekend." ๏ (^) True or false, and why?

Uniform probability distribution

๏ (^) If all outcomes of a sample space are equally likely, then the distribution of probabilities is uniform. ๏ (^) Often when we refer to a “random” choice, we are making the assuming that the outcomes are equally likely. ๏ (^) eg. “pick a random from 1 to 10 at random”, we mean pick a number assuming that each number is equally likely. ๏ (^) Note that this is not always the case.

Discrete sample spaces

๏ (^) For a discrete sample space, ๏ (^) S = {s 1 , s 2 , s 3 , … , sn} ๏ (^) i.e. not infinitely many outcomes, the probability of each outcome can be assigned: ๏ (^) eg. p(sk) = pk ๏ (^) The pk’s form a probability distribution on S. ๏ (^) Example 1.11: Consider the experiment of flipping a fair coin twice and counting the number of heads. We can take the sample space S = { HH, HT, TH, TT }, and let p 1 = … = p 4 = 1/4. ๏ (^) Alternatively, since all we are interested in is the number of heads, we can use the sample space S = {0,1,2}, and let p 0 = 1/4, p 1 = 1/2, p 2 = 1/4.

Conditional probability

๏ (^) Let A and B be two events, with an overlap. ๏ (^) If we know B has occurred, what is the probability that A has also occurred? ๏ (^) The fraction of area of A inside B is ๏ (^) Exercise: Show that p(B|B) = 1. p(A \ B) p(B)

Conditional probability example

๏ (^) Consider two events, A and B: ๏ (^) A = “person who lives more than 80 years” ๏ (^) B = “person who lives more than 81 years” ๏ (^) p(A) = 0.5, p(B) = 0.49, ๏ (^) What is p(B|A)?