STA258 Lecture 01

Pre-Lecture

1. Introduction.pdf

• Basics
  • Definition of Statistics
  • Types of Data
  • Types of Studies
  • Inference
  • Sample Data

STA258 Pre-Lecture Summary 01

Lecture

1. Introduction.pdf

• Mathematical Statistics with Applications (7thEdition) Wackerly, Mendenhall and Scheaffer Cengage Learning; 7th edition
• In a typical Statistics problem we have: a random variable, whose distribution is known. But its parameters are unknown.
• Ex:
  • Lifetime of a rat in a lab, what's suitable? Exponential Distribution. Because the chance for it to live forever is not really possible. It has to die sometime.
  • $X\sim \mathrm{Exp}(\lambda )$
  • What's the value of $\lambda$?
  • We have to infer based on the data we collect, we can get what $\lambda$ is.
  • Or we see something else.
  • $X\sim N(\mu ,{\sigma }^2)$
    • Central Limit Theorem
  • The errors of measurement follow a gaussian / Normal Distribution
  • $X\sim \mathrm{Poisson}(\lambda )$
  • We represent the parameters of our distributions with $\theta$.
  • Instead of $\lambda ,\mu ,{\sigma }^2$ for these distributions, we just use $\theta$
  • So we have $X\sim \mathrm{Exp}(\theta )$ or $X\sim N({\theta }_1,{\theta }_2)$
  • Or $X\sim \mathrm{Poisson}(\theta )$
• How can we estimate the unknown $\theta$ based on our observations.
• Statistic
• Ex:
  • We have $X_1,\dots ,X_n\sim f(X;\theta )$
  • This distribution is parameterized by $\theta$
  • The sequence of Random Variables: $X_1,\dots ,X_n$ is called a sample
  • Each of the RVs is called an Observation
  • Once the sample is Realized, we denote the numerical values taken by each observation using lowercase letters.
    • $x_1,x_2,\dots ,x_n$
  • $X_1\sim \mathrm{Exp}(\lambda )$
  • $X_2\sim \mathrm{Exp}(\lambda )$
  • $X_3\sim \mathrm{Exp}(\lambda )$
  • $\overline{X}$ is an Estimator for $\lambda$
    • Sample mean
    • Sample Mean and Sample Variance#Sample Mean
  • $\overline{x}$ is an Estimate for $\lambda$
    • Our Estimate is always known.
    • Realized sample mean: average
  • Prior to measurement
  • Then we realize it and get our observation:
    • $\begin{array}{cc}\text{Prior to Measurement}& \text{Observations}\\ X_1\sim \mathrm{Exp}(\lambda )& X_1=1\\ X_2\sim \mathrm{Exp}(\lambda )& X_2=2\\ X_3\sim \mathrm{Exp}(\lambda )& X_3=3\\ \text{Probability Theory}& \text{Statistics}\end{array}$
• Note a Statistic is always a function of the sample
  • $\mathrm{\Gamma }=g(X_1,\dots ,X_n)$
  • So $\overline{X}=\frac{X_1+X_2+\cdots +X_n}{n}$ is the sample mean
  • $X_{(n)}=max\{X_1,\dots ,X_n\}$ is the sample max
• Each of these Statistics can be an Estimator
• While we call $T$ an Estimator of $\theta$, we call its realization $t$ an estimate of $\theta$.
• The Estimator is always a random variable
• A sample is random if the observations that you have are iid.
  • $X_1,\dots ,X_n\stackrel{\text{iid}}{\sim }f(x;\theta )$
• We have quantitative and qualitative data.
• Population Studies
• Types of Studies
• Ex:
  • $5$ observations
  • $\begin{array}{c}110\\ 102\\ 130\\ 130\\ 115\end{array}$
  • We can get the sample mean $\overline{X}=\cdots =117.4$
  • We can get variance as $S^2=\frac{1}{n-1}\sum _{i=1}^n(X_i-\overline{X}{)}^2$
  • $S^2=\frac{1}{5-1}[(110-117.4{)}^2+(102-117.4{)}^2+\cdots +(115-117.4{)}^2]=153.4$
  • This is an Estimator for population mean, variance, etc.
• Average is not a robust Estimator:
  • $\begin{array}{c}110\\ 102\\ 130\\ 1000\\ 115\end{array}$
  • We have one outlier
  • Can be due to manual data entry errors or whatever
  • Now average $\overline{X}=\frac{110+102+130+1000+115}{5}=\frac{1457}{5}=291.4$
  • Crazy difference between this and last set.
  • So $\overline{X}$ our sample mean, is not a robust Estimator against outliers.
• Sample Standard Deviation
  • $S=\sqrt{S^2}=\sqrt{153.4}=12.3854753643128$
• Median
  • Sort the data:
    • $\begin{array}{ccccc}102& 120& \underset{\text{Median}}{\underbrace{115}}& 130& 130\end{array}$
  • Median
  • With a data set with outliers
    • $\begin{array}{ccccc}102& 120& \underset{\text{Median}}{\underbrace{115}}& 130& 1000\end{array}$
    • Same Median
    • So this is a robust Estimator against outliers
• Mode
  • Ex:
    • $\begin{array}{ccccc}102& 110& 115& 130& 130\end{array}$
    • See that $130\times 2$ so that's our mode
  • Sometimes it can show that maximum of the distribution, if you have a pdf like normal. Then that's our $\mu$, so we have one parameter.
  • If a distribution has two modes, it's bimodal
  • Average doesn't tell you much here for the whole class.
  • If we have a unimodel distribution, the average does tell us about the class.
  • Otherwise average doesn't mean much.
  • Ex: $\begin{array}{ccccc}50& 60& 70& 80& 80\end{array}$ and $\begin{array}{ccccc}68& 69& 70& 71& 72\end{array}$
  • We have ${\mu }_1=70={\mu }_2$
  • Average doesn't tell us anything, same for both
  • But second set performed more consistent
  • ${\mathrm{range}}_1=40$
  • ${\mathrm{range}}_2=4$