2018-09-20

## Project

• I have a secret… …your project might not work.
• That is okay. Prove to me and to your classmates that:
• You thoroughly understand the substantive area and problem
• You thoroughly understand the data
• You know what methods are reasonable to try and why
• You tried several and evaluated them rigorously, but your predictions are just not that good.
• You can’t get blood from a turnip. (But demonstrate that as best you can.)

## Where did the data come from?

• One row is an observation. What does that mean?
• How are rows generated?

## Replicates

• Common assumption is that data consists of replicates that are “the same.”
• Come from “the same population”
• Come from “the same process”
• The goal of data analysis is to understand what the data tell us about the population.

## Randomness

We often assume that we can treat items as if they were distributed “randomly.”

• That’s so random!
• Result of a coin flip is “random”
• Passengers were screened “at random”
• “random” does not mean “uniform”
• Mathematical formalism: events and probability

## Sample Spaces and Events

• Sample space $$\mathcal{S}$$ is the set of all possible events we might observe. Depends on context.
• Coin flips: $$\mathcal{S}= \{ h, t \}$$
• Eruption times: $$\mathcal{S}= \mathbb{R}^{\ge 0}$$
• (Eruption times, Eruption waits): $$\mathcal{S}= \mathbb{R}^{\ge 0} \times \mathbb{R}^{\ge 0}$$
• An event is a subset of the sample space.
• Observe heads: $$\{ h \}$$
• Observe eruption for 2 minutes: $$\{ 2.0 \}$$
• Observe eruption with length between 1 and 2 minutes and wait between 50 and 70 minutes: $$[1,2] \times [50,70]$$.

## Event Probabilities

Any event can be assigned a probability between $$0$$ and $$1$$ (inclusive).

• $$\Pr(\{h\}) = 0.5$$
• $$\Pr([1,2] \times [50,70]) = 0.10$$

Probability of the observation falling somewhere in the sample space is 1.0.

• $$\Pr(\mathcal{S}) = 1$$

## Interpreting probability: Objectivist view

• Suppose we observe $$n$$ replications of an experiment.
• Let $$n(A)$$ be the number of times event $$A$$ was observed
• $$\lim_{n \to \infty} \frac{n(A)}{n} = \Pr(A)$$
• This is (loosely) Borel’s Law of Large Numbers

• Subjective interpretation is possible as well. (“Bayesian” statistics is related to this idea – more later.)

## Abstraction of data-generating process: Random Variable

• We often reduce data to numbers.
• “$$1$$ means heads, $$0$$ means tails.”
• A random variable is a mapping from the event space to a number (or vector.)

• Usually rendered in uppercase italics

• $$X$$ is every statistician’s favourite, followed closely by $$Y$$ and $$Z$$.

• “Realizations” of $$X$$ are written in lower case, e.g. $$x_1$$, $$x_2$$, …

• We will write the set of possible realizations as: $$\mathcal{X}$$ for $$X$$, $$\mathcal{Y}$$ for $$Y$$, and so on.

## Distributions of random variables

• Realizations are observed according to probabilities specified by the distribution of $$X$$

• Can think of $$X$$ as an “infinite supply of data”

• Separate realizations of the same r.v. $$X$$ are “independent and identically distributed” (i.i.d.)

• Formal definition of a random variable requires measure theory, not covered here

## Probabilities for random variables

Random variable $$X$$, realization $$x$$.

• What is the probability we see $$x$$?
• $$\Pr(X=x)$$, (if lazy, $$\Pr(x)$$, but don’t do this)
• Subsets of the domain of a random variable correspond to events.
• $$\Pr(X > 0)$$ probability that I see a realization that is positive.

## Discrete Random Variables

• Discrete random variables take values from a countable set
• Coin flip $$X$$
• $$\mathcal{X} = \{0,1\}$$
• Number of snowflakes that fall in a day $$Y$$
• $$\mathcal{Y} = \{0, 1, 2, ...\}$$

## Probability Mass Function (PMF)

• For a discrete $$X$$, $$p_{X}(x)$$ gives $$\Pr(X = x)$$.
• Requirement: $$\sum_{x \in \mathcal{X}} p_{X}(x) = 1$$.
• Note that the sum can have an infinite number of terms.

## Probability Mass Function (PMF) Example

$$X$$ is number of “heads” in 20 flips of a fair coin
$$\mathcal{X} = \{0,1,...,20\}$$

## Cumulative Distribution Function (CDF)

• For a discrete $$X$$, $$P_{X}(x)$$ gives $$\Pr(X \le x)$$.
• Requirements:
• $$P$$ is nondecreasing
• $$\sup_{x \in \mathcal{X}} P_{X}(x) = 1$$
• Note:
• $$P_X(b) = \sum_{x \le b} p_X(x)$$
• $$\Pr(a < X \le b) = P_X(b) - P_X(a)$$

## Cumulative Distribution Function (CDF) Example

$$X$$ is number of “heads” in 20 flips of a fair coin

## Continuous random variables

• Continuous random variables take values in intervals of $$\mathbb{R}$$
• Mass $$M$$ of a star
• $$\mathcal{M} = (0,\infty)$$
• Oxygen saturation $$S$$ of blood
• $$\mathcal{S} = [0,1]$$

• For a continuous r.v. $$X$$, $$\Pr(X = x) = 0$$ for all $$x$$.
There is no probability mass function.
• However, $$\Pr(X \in (a,b)) \ne 0$$ in general.

## Probability Density Function (PDF)

• For continuous $$X$$, $$\Pr(X = x) = 0$$ and PMF does not exist.
• However, we define the Probability Density Function $$f_X$$:
• $$\Pr(a \le X \le b) = \int_{a}^{b} f_X(x) \mathrm{\,d}x$$
• Requirement:
• $$\forall x \;f_X(x) > 0$$, $$\int_{-\infty}^\infty f_X(x) \mathrm{\,d}x = 1$$