12 Day 12

Announcements

  • Midterms looked good

    • Scores haven’t been posted yet

    • That is entirely out of my control

    • I will not expedite that process because I am physically incapable of doing that


  • THIS WAS THE EASY EXAM

    • If you found the exam trivial, don’t resolve to stop attending class

      • The next exam will ruin you if you don’t come to class and learn the content

      • Think of class time as a version of study time

        • The probability that your study time is more efficient than class time is low

        • Source: I’ve been doing this (being a student) longer than you


  • If you found the exam difficult/brutal, start attending office hours and help lab

    • The content we cover from here on out is not nearly as intuitive

    • Some would argue portions of it are counter-intuitive

      • \(\text{Let} \ Some=\{Me,Myself, I\}\)


  • I’ve resolved to go against your wishes and paraphrase some of this content

    • All paraphrasing is pre-reviewed/approved by the course coordinator

    • We will hopefully avoid issues of some of the more poorly constructed lessons by doing this


Random Variables

Random variable (shorthand: r.v.):

  • A rule for assigning a numerical value to each outcome of a random experiment

  • General convention is to use a captial letter toward the end of the alphabet to notate these

    • i.e. \(X,Y,Z\)


Flip a fair coin \(3\) independent times

Let r.v. \(X=\{\text{the number of tails observed}\}\)

\[S=\{HHH,HHT,HTH,THH,HTT,THT,TTH,TTT\}\]


  • The r.v. has its own sample space or support

    • Support the set of possible values it can be

  • The support of \(X\) is \(S_X=\{0,1,2,3\}\)

    • Notation clarification - \(S\) generally refers to a sample space, while \(S_N\) where \(N \rightarrow \text{some random variable}\) refers to the support of \(N\)


Use a lowercase letter to represent the observed value of the r.v. so that:

\[x=\text{the value after the experiement has been performed (not random)}\]

\[X=\text{the value before the experiement has been performed (still random)}\]

Why does this notation distinction matter?

  • Statistics is a complete and unique language

    • A troublesome language for first time learners

    • As you come to understand the language better, the rationale for why we use it begins to make more and more sense


In this case we want to make the following expressions make sense:

\[P(X=x)\ \text{means the probability that r.v.} \ X \ \text{is equal to possible value} \ x\]

\[P(X>x) \ \text{means the probability that r.v.} \ X \ \text{is greater than possible value} \ x\]

Random Variable Types

Discrete

  • The number of possible values in the support is finite or countably infinite


Finite or countably infinite refers to integers or whole numbers

  • Let \(X=\{\text{The number of students missing from the class roster today}\}\)

    • The support of \(X\), \(S_X=\{0,1,2,...,40\}\) (assuming my class roster is correct)

    • I can’t take on partial values between values \(1\) and \(2\)

    • Hypothetically I can’t have more than \(40\) students missing on any given day

    • This is Finite and discrete random variable

  • Let \(Y=\{\text{The number of fish in a pond}\}\)

    • The support of \(Y\), \(S_Y=\{0,1,2,...\}\)

    • Can I have half a fish? Technically yes. But half a fish isn’t biologically logical.

    • If partial counts feel arbitrary then you’re more than likely working with a discrete variable

    • I can hypothetically have infinite fish, but the realized value will always be a whole number


Continuous

  • The support consists of all numbers in an interval of the real number line

    • This can be any interval or the entire line

  • There are too many numbers to count (hence: uncountably infinite)


  • Let \(Z=\{\text{The change in median housing prices from one year to another}\}\)

    • The possible values of \(Z\) are: \(-\infty < Z < \infty\)

    • We all know this is strictly positive and every increasing, but let’s pretend

    • Prices can go down, they can go up, they could just not change

    • And those prices can be any value, partial or whole, negative or positive

  • Let \(W=\{\text{The proportion of couples receptive to couples therapy}\}\)

    • The possible values of \(W\) are: \(0 \le W \le 1\)

    • But \(W\) can be anything in between \(0\) and \(1\)

    • Say, \(w=0.012843199\) or maybe \(w=0.99999991\)


r.v. Examples

For each random variable, define whether it is Discrete or Continuous (Bonus: Define the support or possible values of the random variable)

\[X=\{\text{The number that comes up on a die}\}\]


\[Y=\{\text{The height of a randomly chosen college student}\}\]


\[Z=\{\text{The amount of electricity used to light a randomly chosen classroom}\}\]


\[W=\{\text{The number of siblings a randomly chosen person has}\}\]


\[T=\{\text{The length of time it takes to travel from a random classroom to Calvin hall}\}\]



Probability Distributions

Welcome to beginning of real statistics


The form of a r.v.’s probability distribution depends on whether it is continuous or discrete

For a discrete random variable the probability distribution is often a list of all possible values the r.v. can take and their corresponding probabilities of occurrence


Discrete probability distributions satisfy the following two properties:

\[i. \ 0 \le P(X=x) \le 1\]

\[ii. \ \sum_x P(X=x)=1\]


Discrete Probability Distribution Example

Let \(X =\) the number of customers in a line at a supermarket express checkout counter. The probability distribution of \(X\) is given as follows:

\[ \begin{array}{|c|c|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline P(X = x) & 0.4 & 0.2 & 0.15 & 0.1 & 0.1 & 0.05 \\ \hline \end{array} \]

Q1. Is this a legitimate probability distribution?


Q2. Find the probability that there is no customer in a line at the express checkout.


Q3. Find the probability that there is at least one customer in a line at the express checkout.


Q4. \(P(2 < X \le 5)\)



Probability Distributions (Continued)

For a continuous random variable probabilities are determined via a probability density function or pdf

  • To actually dig into this discussion properly, we need to use Calculus

  • The remainder of today will just focus on discrete concepts


But making an aside that you shouldn’t take notes on

  • The calculus you’d take through the mathematics department (Calc 1, 2, and 3) is not necessary or even all that useful for applied statistics

  • In actuality, applied statisticians use something called numerical calculus to perform their computations

    • This roughly means we’re taking approximations of the values we’d use traditional calculus to realize analytically

    • This also means we can do things you absolutely cannot with pencil and paper mathematics

  • If the thing that bars you from diving into the subject of statistics further is the fear of higher mathematics like Calculus and Differential Equations

    • You don’t need to be good at those things, just understand their concepts


Getting back to the lesson


We can draw a histogram so that the area of each bar above a given possible value of a r.v. is equal to it’s probability of occurrence

As with datasets, probability distributions can have shape and measures of center as well as spread

  • What is the shape of this probability distribution?

Mean of a Discrete Probability Distribution

Mathematicians (and Statisticians) prefer deriving values and writing equations in more general forms

  • Not because we like to confuse people, but because we’re lazy

We want to develop a general form for a measure of center and variability for a discrete probability distribution


  • Consider the probability distribution of \(X\) to be a population

  • Numerical values that we calculate from a probability distribution are called parameters

  • We would notate the mean of a population with \(\mu\)

    • We can use the same notation for the mean of a probability distribution


Given several random variables to consider, we can denote the mean of r.v. \(X\) as:

\[\mu X \ \ \ \ \ \text{or} \ \ \ \ \ E(X)\]

For a discrete probability distribution of r.v. \(X\), the mean is given by:

\[E(X)=\mu=\sum_x xP(X=x)\]

  • We call this “the weighted sum of all probabilities of \(x\)”

Mean of X Example

\(X=\{\text{number of customers in a line at the express checkout counter}\}\)

\[ \begin{array}{|c|c|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline P(X = x) & 0.4 & 0.2 & 0.15 & 0.1 & 0.1 & 0.05 \\ \hline \end{array} \]

Recalling our definition of the mean of a probability distribution for a discrete r.v.

\[E(X)=\mu=\sum_x xP(X=x)\]

We get:

\[\mu_X=\sum_x xP(X=x)\] \[=0(0.4)+1(0.2)+2(0.15)+3(0.1)+4(0.1)+5(0.05)=1.45\]

  • So we would say: “over time we expect to have \(1.45\) customers in a line at the express checkout counter”


Interpretation of the Mean of a r.v.

We’ve previously discussed how the mean of a dataset is it’s balance point (fulcrum)

  • Shockingly, the exact same interpretation applies

You can also consider \(\mu\) to be the “average in the long run”

  • Using the Law of Large Numbers:

    • As we produce more observations and take their averages (\(\bar{x}\))

    • We should approach/converge on the value of \(\mu\)


Variance of a Discrete Probability Distribution

As discussed, probability distributions have spread, spread we can measure

Denoting the variance of discrete r.v. \(X\):

\[\sigma^2_X \ \ \ \ \ \text{or} \ \ \ \ \ Var(X)\]

The general formula for the variance of r.v. \(X\):

\[\sigma^2=\sum_x (x-\mu)^2 P(X=x)\]


Variance of X Example

\(X=\{\text{number of customers in a line at the express checkout counter}\}\)

\[ \begin{array}{|c|c|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline P(X = x) & 0.4 & 0.2 & 0.15 & 0.1 & 0.1 & 0.05 \\ \hline \end{array} \]

Recalling our definition of variance:

\[\sigma^2=\sum_x (x-\mu)^2 P(X=x)\]

We get:

\[\sigma^2_X=\sum_x (x-\mu)^2 P(X=x), \ \ \ \text{recall} \ \mu=1.45\]

\[=(0-1.45)^2(0.4)+(1-1.45)^2(0.2)+(2-1.45)^2(0.15)+(3-1.45)^2(0.1)+(4-1.45)^2(0.1)+(5-1.45)^2(0.05)\]

\[=2.4475\]


For standard deviation, defined loosely as the “average” distance from \(\mu\) in the probability distribution:

\[\sigma=\sqrt{\sigma^2}\]

So:

\[\sigma_X=\sqrt{2.4475}\approx 1.564\]

In-class Exercise

Given the following probability distribution of discrete r.v. \(X\):

\[ \begin{array}{|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 \\ \hline P(X = x) & 0.3 & 0.4 & 0.2 & 0.1 \\ \hline \end{array} \]

  1. Is this a proper probability distribution? Why or why not?



  1. Find \(E(X)\)



  1. Find \(Var(X)\)



  1. Find \(\sigma_X\)




Given the probability histogram for r.v. \(X\):

  • What is it’s shape?


  • Ok go away