STAT 206 - Definitions

Not a comprehensive list

Sec 1.1 §

• Probability - Models uncertainty and variability

Defining Probability: §

• Experiment - A situation where chance or uncertainty leads to results (aka outcomes)
• Outcome - Result of a single trial of an experiment
• Event - One or more outcomes of an experiment
• Sample Space (S) - All possible distinct outcomes in an experiment

Classical definition of probability: §

$\frac{\text{\# of the ways the event can occur}}{\text{\# of outcomes in S}}$

Disadvantages of Classical definition

• Sample Space S needs to be finite
• Outcomes in S need to be equally likely
• Outcomes in S (or event A) may be difficult to count

Relative Frequency Definition §

Probabilities are assigned on the basis of experimentation or historical data

The probability of an event is the (limiting) proportion (or fraction) of times the event occurs when the experiment is repeated a large number of times under the exact same conditions.

Disadvantages of Frequency Definition

Need an infinite # of experiments to get the correct value. Sometimes the recreation of the experiment under the same controlled conditions may be challenging

Subjective Probability Definition §

The probability of an event is based on how confident the person making the statement is that the event will occur. Usually based on prior knowledge (belief) or available information

Disadvantages of Frequency Definition

• No mathematical model is used
• How do you determine who’s knowledge / judgement is superior

Sec 1.2 §

Sample space is set of distinct outcomes for an experiment/process so in a single trial one and only one of these outcomes can occur

$S=1,2,3,4,5,6,...$

Only one of these outcomes can occur

Sample space is not necessarily unique

confused so asked gpt

Here, “not necessarily unique” refers to the fact that the choice of sample space is not one-of-a-kind — there can be more than one valid way to represent it.

• Example: For the same die roll, we might define the sample space as:
  S1={1,2,3,4,5,6}S_1 = {1, 2, 3, 4, 5, 6}S1​={1,2,3,4,5,6}
  or, alternatively,
  S2={odd,even}S_2 = {\text{odd}, \text{even}}S2​={odd,even}.

Both are legitimate sample spaces, just framed differently. Neither is “unique” in the sense of being the only correct version.

So “not necessarily unique” = different valid representations are possible.

Discrete sample space is one that consists of finite or countable infinite set of outcomes

In discrete sample spaces, we can talk about:

• Simple Event (Outcome) - An event that contains only one point
• Compound Event - An event made up of 2 or more simple events

Probability Laws §

$S=\{a_1,a_2,a_3,\dots \}$, discrete sample space

Probabilities $P(a_i)$ for $i=1,2,3,\dots$ must satisfy 2 conditions

1. $0\le P(a_i)\le 1$
2. ${\sum }_{\text{all i}}P(a_i)=1$

Probability P(A) of an event A is defined as

$P(A)={\sum }_{a\in A}P(a)$

Odds §

Term odds can be used to describe probabilities

odds in favor of event A occurring is given by:

$P(A):1-P(A)\text{or}\frac{P(A)}{1-P(A)}$

Odds against the event A is the ratio

$1-P(A):P(A)or\frac{1-P(A)}{P(A)}$

Sec 1.3 §

Addition & Multiplication Rule §

Job 1 can be done in p ways

Job 2 can be done in q ways

We can do either job 1 or job 2 in p + q ways

OR implies addition

Similarly

AND implies multiplication

Sampling w/ and w/o replacement §

With replacement means that every time an object is selected it’s put back into the pool

Without replacement means that every time an object is selected it is NOT put back

Permutations - Arrangement of objects where order matters

STAT 231 §

Introduction to Statistical Science §

• Population - Population is a collection of units, unit as individual person, place, or thing about which we can take measurements
• Process - Process is a collection of units, but those units are ‘produced’ over time
• Variates - Studying specifics about units, eg: for dogs breed of dog, weight of dog, general health, etc.
• Continuous - Variates that can be measured to an infinite degree of accuracy
• Discrete - Variates that can only take a finite or countable infinite number of values
• Categorical - Variates in which units fall into a non-numeric category such as hair color, health of dogs, etc.
• Ordinal - Categorical variates where an ordering is implied
• Complex - More unusual variates such such open-ended responses to survey question, or image
• Sample Variance -

$$s^2=\frac{1}{(n-1)}\sum _{i=1}^n(y_i-\tilde{y}{)}^2$$

$=$

$$\frac{1}{(n-1)}[\sum _{i=1}^n(y_i^2)-n(\tilde{y}{)}^2]$$

• Sample Standard Deviation - Denoted by s and is just the square root of $s^2$
• Define the $p^{th}$ quantile (0 < p < 1)
  • Let m = (n + 1)p
  • $1\le m\le n$ then take the $m^{th}$ smallest value $q(p)=y_{(m)}$
  • If $m$ is not an integer, but $1<m<n$, then determine the closest integer $j$ s.t. $j<m<j+1$ and take $q(p)=\frac{1}{2}[y_{(j)}+y_{(j+1)}]$
• Interquartile Range - The difference between the upper and lower quartiles = q(0.75) - q(0.25)
• Negatively Skewed - Mean is less than the median
• Positively Skewed - Mean is greater than the median
• Symmetry = Skewness of 0
• Sample Skewness -

$$g_1=\frac{\frac{1}{n}{\sum }_{i=1}^n{\left(y_i-\bar{y}\right)}^3}{{\left[\frac{1}{n}{\sum }_{i=1}^n{\left(y_i-\bar{y}\right)}^2\right]}^{3/2}}$$

- Numerator indicates sign for skewness coefficient, denom is always positive, more skewed the distribution is the higher the abs value of $g_1$ will be

• Kurtosis - Measures the heaviness of the tails and peakedness of the distribution of dat
• Sample Kurtosis -

$$g_2=\frac{\frac{1}{n}{\sum }_{i=1}^n{\left(y_i-\bar{y}\right)}^4}{{\left[\frac{1}{n}{\sum }_{i=1}^n{\left(y_i-\bar{y}\right)}^2\right]}^2}$$

- Sample Kurtosis is always positive and doesn't have units
- Kurtosis > 3 = heavier tails, more peaked center
- Kurtosis < 3 = lighter to no tails and a more flat centre

• Maximum Likelihood Estimate §