Motivation

Describe the support of the following random variables:

\[ \begin{array}{|c|c|} \hline \text{r.v.} & \text{Description} \\ \hline T & \text{Time it takes for a drug to enter the blood stream} \\ \hline U & \text{Probability of landing on 3 on a fair 6-sided dice} \\ \hline V & \text{Volume of the water in a swimming pool} \\ \hline W & \text{Proportion of people who get sick at an Aggieville bar}\\ \hline X & \text{The result of 50 coin flips} \\ \hline Y & \text{Number of wild boar observed in a 10 acre plot} \\ \hline \end{array} \]

Normality

Given:

\[ X \sim N(\mu,\sigma^2) \]

• \(X\) is continuous

• The support of \(X\) is \(S_X = (-\infty,\infty)\)

• \(X\) has a centered mean as well as constant variance

  • Both are finite (a.k.a., they exist)

Design vs. Modeling

There are two ways to approach the application of statistics: Design or modeling

• We can use simplistic modeling if we have a very good study/experimental design

• If our design is flawed we can make up for this with better modeling

• Both can be improved with increased sample sizes

Modeling

For modeling we have two options for getting stronger inference:

• Increase sample size

• Load on assumptions

  • Is the assumption of normality asking much?

  • If \(\mu\) is large and \(\sigma\) is small we don’t care about the lower bounds

Continuous

\[ \begin{aligned} & Y = \{\text{The number of wild boar observed in a 10 acre plot}\} \\ & S_Y = \{0,1,2,...\} \\ \end{aligned} \]

Let \(\ell = \ln{(Y)}\)

\[ S_{\ell} = [0,\infty) \]

Negative support

\[ T = \{\text{The time it takes for a drug to enter the blood stream}\} \]

Let \(\tilde T = T + 100\)

• Now no matter how small \(T\) is, \(\tilde T\) won’t cross \(0\)

• If \(\tilde T\) looks bell shaped we can use the normal distribution!

Symmetry

\[ W = \{\text{The proportion of people who get sick at an Aggieville bar}\} \\ \]

Say that \(W\) was right-skewed:

• Let \(Q = \arcsin \sqrt{W}\)

• the arcsine is asinine

Symmetry

Assumptions

• Hand-wavey transformations are useful

  • But they can be uncomfortable

• Let’s look at what we dealt with last class

\[ \mu_{\bar x} = \mu \quad , \quad \sigma_{\bar x} = \frac{\sigma}{\sqrt{n}} \]

Bias vs. Variance

Let \(\mu = 10\) and \(\sigma = 5\). Show the distribution of \(\bar x\) at \(n=1\), \(n=10\), \(n=100\), and \(n=1000\).

Assumptions = Samples?

As we increase assumptions we see the same result:

• More assumptions \(\rightarrow\) more bias

• Less assumptions \(\rightarrow\) less variance

What’s a possible explanation for this? Does the opposite relationship make sense?

Distribution Theory

• In distribution theory we seek to:

  • Formally describe distributions of random variables

  • Determine their moments and other features

  • Put them into practice

  • Locate any useful transformations or cases of the distributions

Bernoulli

Let \(X\) be the result of a fair coin toss.

\[ X = \begin{cases} 1 & \text{if heads} \\ 0 & \text{if tails} \\ \end{cases} \]

\[ P(X=x)= \begin{cases} 0.5 & \text{if }\ x=1 \\ 0.5 & \text{if }\ x=0 \end{cases} \]

Bernoulli

Let \(X\) be the result of an unfair coin toss.

• We don’t know the probability of heads, let’s label it \(p\)

\[ P(X=x)= \begin{cases} p & \text{if }\ x=1 \\ 1-p & \text{if }\ x=0 \end{cases} \]

\[X \sim \text{Bern}(p)\]

Binomial

Let \(X\) be the result of \(2\) fair coin tosses.

• There are multiple combinations of \(X=x\):

\[ \begin{array}{|c|c|} \hline \text{Flip 1} & \text{Flip 2}\\ \hline \text{1} & \text{1}\\ \hline \text{1} & \text{0}\\ \hline \text{0} & \text{1}\\ \hline \text{0} & \text{0}\\ \hline \end{array} \]

Binomial

Thinking a little differently:

• We have \(3\) possible outcomes (if order doesn’t matter)

\[ \begin{array}{|c|c|c|c|} \hline & 0 & 1 & 2\\ \hline 0 & 1 & 0 & 0 \\ \hline 1 & 1 & 1 & 0 \\ \hline 2 & 1 & 2 & 1 \\ \hline \end{array} \]

Binomial

We can represent this with the “choose” function:

\[ {n \choose x} = \frac{n!}{x!(n-x)!} \]

\(\text{where } n!=n\times(n-1)\times (n-2) \times...\times3 \times 2 \times 1\)

If the flips are independent we can express the probability as:

\[ \begin{aligned} & \text{Success} = p^x \\ & \text{Failure} = (1-p)^{n-x} \\ \end{aligned} \]

Binomial

Smashing those together:

\[P(X=x)={n \choose x}p^x(1-p)^{n-x}\]

\[X \sim \text{Binom}(n,p)\]

Binomial

\[ \begin{aligned} & X \sim \text{Binom}(n,p) \\ \\ & \mathbb{E}X = np \\ \\ & \mathbb{V}X=np(1-p) \\ \\ \end{aligned} \]

• What about Bernoulli?

Poisson

Poisson: the French word for “fish”

• Think about counting fish in a pond

  • Discrete and strictly positive

\[X \sim \text{Pois}(\lambda)\]

\[\mathbb{E}X=\mathbb{V}X=\lambda\]

Poisson

Uniform

• Before we defined this as a shape

• Imagine now that you can control the interval of that uniform bar

\[f(x)=\begin{cases} \frac{1}{b-a} & \text{for }\ a \leq x \leq b \\ 0 & \text{otherwise} \end{cases}\]

\[X\sim \text{Unif}(a,b)\]

Beta

• For data that takes on values between \(0\) and \(1\)

  • And those values are not constant

\[X \sim \text{Beta}(\alpha,\beta)\]

\[\mathbb{E}X=\frac{\alpha}{\alpha+\beta} \quad \quad \mathbb{V}X=\frac{\alpha \beta}{(\alpha+ \beta)^2(\alpha+\beta+1)}\]

Beta

Mixing Distributions

Time

Is time discrete or continuous?

• Imagine how you would describe both!

• Now imagine the constraints for both

  • Can time be negative?

  • Can time stretch to infinities?

Gamma

For continuous and strictly positive data:

\[X \sim \text{Gamma}(\alpha,\beta)\]

• The expectation and variance of this differ

  • Depends how we parameterize it

  • As such, we won’t discuss those

Gamma

Models

• Any mathematical representation of a process

• Think about model planes:

  • Can we make a very simple one?

  • What about a very complex one?

  • Can they both fly?

  • Is either as good as a real plane?

Phenomenological Models

Building a model, blind of a process, to better understand a process.

\[y = mx + b\]

Mechanistic Models

Building a model using knowledge of a process, to better understand the data.

\[\lambda(t)=\lambda_0e^{\gamma(t-t_0)}\]

Linear

Linear in the parameters:

\[Y=aX+b\]

\[w=\beta_0+\beta_1x+\beta_2x^2\]

\[P=\mu+\alpha\log(r\times t)\]

\[\boldsymbol \Delta=\boldsymbol \beta \cos(\boldsymbol X)+10\]

Nonlinear

We did something nonlinear to those parameters:

\[Y=a^2X+b\]

\[w=\beta_0+\log(\beta_1)x\]

\[\lambda(t)=\lambda_0e^{\gamma(t-t_0)}\]

Deterministic

The same inputs always result in the same outputs:

\[y = 2x\]

• Let \(x=4\)

• Let \(x=4\) again

• Did you get the same result?

Probabilistic

There’s some random element to the model:

\[ y = mx + b + \text{Random Error} \]

• The same input won’t always get the same output

  • This is where we can slap distributions into the mix

“Solving” models

• Explicit models: Input values for the parameters and solve for the output (a.k.a. response)

• Implicit models: Solve for the parameters to understand the structure of the model

• We do both in statistics (usually back to back)

\[P(t)=P_0e^{rt}\]

\[y=mx+b\]

Derived Quantities

Final Note

There’s two more “distinctions” of models, specific to statistics:

• Frequentist

  • This is every method we learn in this class

• Bayesian

  • This exists, we’ll talk about it a little, not too much

  • STAT 341, STAT 610/611, STAT 768