# Lecture_5
Author: Zheyuan Wu
Date: 2023-01-25

#### Zheyuan Wu

#### 2023-01-25

> first exam cover chapter two.

## Mean and Will Rogers Paradox

> “There are three kinds of lies: lies, damned lies, and statistics.”

When we change an observation (eg.4) from Group A to Group B, both mean increased.

### Save life without treatment.

    # Life span of two groups
    Healthy=c(60,70,80,90,100)
    Cancer=c(45,50,55,60,65)
    mean(Healthy)

    ## [1] 80

    mean(Cancer)

    ## [1] 55

By changing diagnose method or imaging techniques.

    # Change the 60 to the other group
    Healthy=c(70,80,90,100)
    Cancer=c(45,50,55,60,60,65)
    mean(Healthy)

    ## [1] 85

    mean(Cancer)

    ## [1] 55.83333

Mean is sensitive to outlier and may not represent a “typical” value of the group.

## Introduction to Probability

### Experiments, outcomes, sample space, and events

Event: Pulling out an orange ball (doesn’t matter which one).

Random draw: Put my hand in the bowl and pull ball out without looking.

record the color and put them back.

Experiment: single, “random” trail

- Pull ball out of bowl
- Flipping coin

Outcomes: observable, potential result of the trial

- Getting a blue ball
- Getting a head

Sample Space: Set of all outcomes.

- set(blue, orange, black, white …)
- set(head, tail)

Event: any subset of the sample space (subset of sample space)

- set(blue)
- set(tail)

Probability of an event: the relative frequency with which that event can be expected to occur

WE could also think of probability as being a function that takes every event and assigned between 1 and 0.

\[ P(X) \]

### Union, Intersection, complement, disjoint Events

Subset: collection of any outcomes

Elementary event: an event consisting of a single outcome

empty set: set that does not have any element.

Complement of event of A are all the space other than event A (denoted by \(A^c\))

Multiple events

Intersection:

\[ C = A \cup B \] Mutrually exclusive event: events that don’t have any overlap (\(C = A \cup B =\phi\)) (independent from each other)

Union:

\[ C = A \cap B \]

Difference and Subset: A-B = the event have the outcomes contains A but not B

if \(A\cup B=A\),then A is a subset of B, denoted by (\(A \subseteq B\))

We may also use this notation for sum in union / intersection operations

\[ \cup^k_{i=1}A_i = A_1\cup A_2 \cup A_3 ... \cup A_k \] Union otherwise..

Union… At least one sample is acceptable

Intersect… All the sample is acceptable

#### Set operations

1. Commutative Laws: \[ A\cup B=B\cup A,A\cap B=B \cap A \]

2. Associative Laws: \[ (A\cup B)\cup C=A\cup(B\cup C),(A\cap B)\cap C=A\cap(B\cap C) \]

3. Distributive Laws: \[ (A\cup B)\cap C=(A\cap C)\cup(B\cap C),(A\cap B)\cup C=(A\cup C)\cap(B\cup C) \]

4. De Morgan’s Laws: \[ (A\cup B)^c=A^c\cap B^c, (A\cap B)^c=A^c\cup B^c \]

### Axioms of probability

We want probability to be relative frequency with which we expect the event to occur

Axiom 1: \(P(S)=1\)

Axiom 2: \(0\leq P(A) \leq 1\)

Axiom 3: \(P(A\cup B)=P(A)+P(B)\) if they are multrally exclusive.