recap: best guesses

Sample statistics are best guesses about characterstics of the population; we estiamte unknown population parameters with sample statistics, respectively.

Distributions, proportions from histogram

We can use a sample (visualized as a histogram) to estimate the proportion of people

• between 180 and 190cm tall, or
• taller than 220cm,
• shorter than 201cm.

Distributions, proprotions from histogram

Proportion of sample between 180 and 190cm tall?

\(\frac{\text{# people between 180 and 190}}{n} =\) 0.3411

Distributions, connecting to histograms

We are still estimating things about the popluation. Now we are estimating proportions between two numbers, instead of where the distribution is centered (\(\mu\)) or how wide the distribution is (\(\sigma\)).

Distributions, in simple language

Distributions are the population analogue to histograms, just as \(\mu\) is the population analogue to \(\bar{X}\). Distributions are mathematical descriptions of populations, defined by a function that has a measure of center \(\mu\) and a measure of spread \(\sigma\).

Distributions, an example

We often model adult male heights with a continuous distribution. Simplified, this means we draw a curve over a histogram with decreasing binwidths of an infinite set heights.

Distributions, an example

Distributions, an example

Distributions, an example

Distributions, an example

Without a sample, we hypothesize the (population) distribution of heights to have a nice mathematical form.

Continuous Distribution

This smooth curve represents a probability density function (also called a density or distribution). The toal area under the density function is equal to 1.

Distributions, probability

What is the probability that a randomly selected adult is between 180 and 190cm tall?

\(P(180 < X < 190) =\) 0.3413

Distributions, sample estimates population

The proportion of people within the sample in between 180cm and 190cm 0.3411 is a close estimate of the population probability that a randomly chosen person is between those same heights 0.3413.

Distributions, proprotions from histogram

Proportion of sample less than 201?

\(\frac{\# \text{people} < 201}{n} =\) 0.8641

Distributions, probability

What is the probability that a randomly selected adult is shorter than 201?

\(P(X < 201) =\) 0.864

Distributions, sample estimates population

Again, the proportion of people within the sample that are shorter than 201cm 0.8641 is a close estimate of the population probability that a randomly chosen person is shorter than \(201\)cm 0.864.

Distributions, population known?

Though, there is something odd going on. We’ve assumed we know the population distribution; it’s shape, it’s mean, and it’s standard deviation.

Distributions, in practice

In practice, we don’t and won’t know the population distribution. Nevertheless, we can learn things about it via a sample. Soon we’ll find out that this lack of knowledge about the population, really just doesn’t matter.

Random Variables, an example

Continuing our height example: in the case of randomly selected individuals and their height, the random variable \(X\) is the height of an unknown (to be randomly selected) individual.

Random Variables, an example

In this example, the random variable is an adult’s height. Since we don’t yet know how tall a to be randomly selected adult is, we denote the to be determined value by \(X\). Once we observe the variable’s outcome, we write the value of the adult as \(x\).

Random Variables, definition

A random process or variable with a numerical outcome, see OS4 Section 3.4.1.

Random Variables, note

Each random variable has a distribution function that describes the form of the random variable. Tying the pieces together, we say that each random variable (via its distribution function) has a measure of center \(\mu\) and a measure of spread \(\sigma\).

Uniform Distribution

The discrete uniform distribution is the simplest of discrete distributions. We write \(X \sim U(a, b)\), meaning the random variable \(X\) is distributed uniformly on the interval \([a, b]\).

Uniform Distribution, example

The canonical example is die rolling. If we let the random process of die rolling be denoted by \(X\), then \(X \sim U(1, 6)\).

Bernoulli Distribution, motivation I

Suppose you work for the IRS and you are interested in the probability that a randomly selected tax payer is attempting to commit fraud on their tax return. So you and your boss decide to test this. From a sample of 900 tax payers, you perform an in depth search through each of the 900 tax forms.

Bernoulli Distribution, motivation III

You are interested in calculating the average G/C fraction of Human genomic DNA across the whole genome. You sample \(50\) individuals, sequence their genome …

Bernoulli Distribution, introduction

Notice what is common to all of these scenarios

• random variable has just two outcomes: “success” or “failure”

  • commit fraud or not, dominant or not, G/C base or not

• independent trials

  • your not paying taxes does not (necessarily) determine your neighbor’s willingness to pay taxes, …, G/C base here does not dictate next base

• probability of “success,” \(p\), stays the same

  • Bill through Ted have 0.62% chance of commiting fraud, …, dominant gene A shows with frequency of 0.1

Bernoulli Distribution, more formally

The Bernoulli distribution describes the probability of seeing a successes, calculated from \(n\) independent trials with probability of a success \(p\).

Bernoulli Distribution, conditions met?

Let’s discuss: suppose you work for the IRS and you are interested in the probability that a randomly selected tax payer is attempting to commit fraud on their tax return. So you and your boss decide to test this. From a sample of 900 tax payers, you perform an in depth search through each of the 900 tax forms.

Binomial Distribution

If we can estimate \(p\) from a sample of Bernoulli random variables, we can then answer questions of the sort

• What is the probability that 3 of the next 9 tax forms are found to be fradulent?
• What is the probability that 20 of 50 randomly sampled individuals will display the dominant phenotype?
• …

Binomial Distribution

Normal Distribution

The normal distribution is ubiquitous in statistics.

Normal Distribution

The normal distribution is a probability density function that is symmetric, unimodal, and bell shaped.

Notes

• area under the curve is equal to 1,
• perfectly symmetric about \(\mu\),
• parameters: centered at \(\mu\) with standard deviation \(\sigma\),
• often \(\mu = 0\), shifts the distribution,
• often \(\sigma = 1\), scales the distribution,
• we write in short hand \(N(\mu, \sigma)\),
• use \(Z\) to denote \(N(0,1)\)
• Gau\(\beta\)ian distribution

Interlude, z-score

The z-score of an observation is the number of standard deviations the observation lies above or below the mean. We compute the z-score for an observation \(x\) that follows a distribution with mean \(\mu\) and standard deviation \(\sigma\) using \[z = \frac{x - \mu}{\sigma}\]

z-score notes

• The Z-score is a unitless number. Why?
• By definition, if an observation is 1 standard deviation above its mean, the z-score is 1. If an observation is 1.5 standard deviations below the mean, its z-score is -1.5.

Z-score, example

• Suppose SAT scores are distributed \(X \sim N(1500, 300)\) and ACT scores are distributed \(N(21, 5)\). If Ann scored 1800 on the SAT and Tom scored 24 on the ACT, who performed better?

• \(z_A = \frac{1800 - 1500}{300} =\) 1

• \(z_T = \frac{24 - 21}{5} =\) 0.6

• These are effectively quantiles/percentiles. Picture?

68, 95, 99.7 Rule

Use R to justify the following approximate numbers.

Random Variables, expected value

All random variables have a theoretical mean, called the expected value. The sample mean \(\bar{X}\) is the sample analogue of this quantity.

• The expected value of \(X\), written \(\mu = E(X)\), is the value we’d expect to get out of \(X\) is we infinitely repeated the process that generated \(X\) and calculated the mean.

Random Variables, expected value example

Let \(Y\) take on the values \(0,1\) with equal probability – think coin flip. We can use R to simulate this random variable. We generate a sequence \(y_1, y_2, ..., y_{10000}\), each of which will be \(0\) or \(1\) once observed, and calculate the running mean. That is,

• \(Y_1\) evaluates to \(y_1\), with mean \(y_1 / 1\)
• \(Y_1, Y_2\) evaluate to \(y_1, y_2\), with mean \((y_1 + y_2)/2\)
• \(Y_1, Y_2, Y_3\) evaluate to \(y_1, y_2, y_3\), with mean \((y_1 + y_2 + y_3)/3\)
• …
• \(Y_1, ..., Y_{10000}\) evaluate to \(y_1, ..., y_{10000}\), with mean \(\sum_{i=1}^{10000} y_i /10000\).

Since we expect the long-run average of a coin flip to be \(1/2\), our running mean should converge to \(1/2\).

Random Variables, variance

Random variables also have a common measure of spread, called the variance. The squared sample standard deviation, \(s^2\), is the sample analogue of this quantity.

• The variance of \(X\), written \(\sigma^2 = Var(X)\), is the mean of squared deviations about \(E(X)\) if we infinitely repeated the process that generated \(X\).

Random Variables, variance example

Consider \(Y\) a random variable following a Bernoulli distribution with probability of “success” \(p\).

Random Variables, distributions

Random variables are said to follow specific distribution functions, eg \(X\) is distributed as [some name].

We calculate the expected value and variance of random variables from these distribution functions. When we calculate statistics, it is these mathematical quantities that we are estimating.

Take Away

• We assume distributions describe populations of interest
  • distributions describe the shape of our data
• Distributions have parameters: mean, variance, percentiles, …
• Use data to estimate parameters
  • use data to learn about population