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MANE 6313

Week 3, Module A

Student Learning Outcome

Analyze simple comparative experiments and experiments with a single factor.

Module Learning Outcome

Review probability distributions.

Simple Comparative Experiments

  • Chapter two only focuses on simple experiments to compare two conditions. Later, we will extend these techniques.

  • Largely review of statistics and hypothesis testing from previous classes.

  • Statistics focuses on two emphases: Descriptive statistics and inferential statistics.

  • In inferential statistics we are concerned with estimation and hypothesis testing.

  • Differences or fluctuations in outcomes from the same treatment are called noise in the results

  • Often this noise is referred to as experimental error or simply error

  • We classify statistical error as those errors that arise from variation that is uncontrolled and generally unavoidable.

  • The presence of noise implies that the response variable is a random variable

Review of Probability Distributions

  • The probability structure of a random variable is defined by its probability distribution.

  • For a discrete random variable

\[ \begin{aligned} 0\leq p(y_i)&\leq& 1\;\mbox{for all values of }y_i\\ P(y=y_i)&=&p(y_i)\;\mbox{for all values of }y_i\\ \sum_{\mbox{all values of }y_i}p(y_i)&=&1 \end{aligned} \]
  • For a continuous random variable
\[ \begin{aligned} f(y)\geq 0\\ P(a\leq y\leq b)&=&\int_a^bf(y)\,dy\\ \int_{-\infty}^{\infty}f(y)\,dy=1 \end{aligned} \]
  • The mean of a probability distribution is defined to be
\[ \mu=\left\{\begin{array}{l l} \int_{-\infty}^{\infty}yf(y)\,dy & y\mbox{ continuous}\\ \sum_{\mbox{all y}}yp(y) & y\mbox{ discrete}\end{array}\right. \]
  • The expected value operator is \(\(E(y)\equiv \mu\)\)
  • The variance of a probability distribution is defined to be
\[ \sigma^2=\left\{\begin{array}{l l} \int_{-\infty}^{\infty}(y-\mu)^2f(y)\,dy & y\mbox{ continuous}\\ \sum_{\mbox{all y}}(y-\mu)^2p(y) & y\mbox{ discrete}\end{array}\right. \]

  • The variance can be written in terms of the expectation \(\sigma^2=E\left[(y-\mu)^2\right]\)

  • We will often use the variance operator \(V(y)\equiv E\left[(y-\mu)^2\right] =\sigma^2\)

Common Probability Distributions1

Common Distributions

Functions of Random Variables

  1. \(E(c)=c\)

  2. \(E(y)=\mu\)

  3. \(E(cy)=cE(y)=c\mu\)

  4. \(V(c)=0\)

  5. \(V(y)=\sigma^2\)

  6. \(V(cy)=c^2V(y)=c^2\sigma^2\)

For two random variables with the following properties \(E(y_1)=\mu_1\), \(V(y_1)=\sigma_1^2\), \(E(y_2)=\mu_2\) and \(V(y_2)=\sigma_2^2\)

  1. \(E(y_1+y_2)=E(y_1)+E(y_2)=\mu_1+\mu_2\)

  2. \(V(y_1+y_2)=V(y_1)+V(y_2)+2Cov(y_1,y_2)\) where \(Cov(y_1,y_2)=E\left[(y_1-\mu_1)(y_2-\mu_2)\right]\)

  3. \(V(y_1-y_2)=V(y_1)+V(y_2)-2Cov(y_1,y_2)\)

If \(y_1\) and \(y_2\) are independent, then

  1. \(V(y_1\pm y_2)=V(y_1)+V(y_2)=\sigma_1^2+\sigma_2^2\)

  2. \(E(y_1\cdot y_2)=E(y_1)\cdot E(y_2)=\mu_1\cdot\mu_2\)

In general note that

\[ E\left(\frac{y_1}{y_2}\right)\neq\frac{E(y_1)}{E(y_2)} \]

  • For two random variables to be independent we need to examine the joint distribution function \(f_{y_1,y_2}(y_1,y_2)\) and the two marginal distribution functions \(f_{y_1}(y_1)\) and \(f_{y_2}(y_2)\)

  • We need to show that \(f_{y_1,y_2}(y_1,y_2)=f_{y_1}(y_1)\cdot f_{y_2}(y_2)\)

  • For normally distributed random variables (and only normally distributed r.v.) if \(Cov(y_1,y_2)=0\) then \(y_1\) and \(y_2\) are independent.

Sampling

  • We will use the following statistics as point estimators for \(\mu\) and \(\sigma^2\) respectively

$$ \overline{y}=\frac{\sum_{i=1}^ny_i}{n} $$ and $$ S^2=\frac{\sum_{i=1}^n(y_i-\overline{y})^2}{n-1} $$ - Properties of estimators

  • Properties of point estimators

    • Unbiased

    • Minimum variance

  • The normal distribution is one of the most important sampling distributions

Central Limit Theorem

If \(y_1,y_2,\ldots,y_n\) is a sequence of \(n\) independent and identically distributed random variables with \(E(y_i)=\mu\) and \(V(y_i)=\sigma^2\) (both finite) and \(x=y_1+y_2+\cdots +y_n\) then $$ z_n=\frac{x-n\mu}{\sqrt{n\sigma^2}} $$ has an approximate \(N(0,1)\) distribution

  • What is the implication here?

CLT Demonstration2

CLT Demonstration

  1. Montgomery and Runger (2014). Applied Statistics and Probability for Engineers, 6th edition. John Wiley & Sons. ↩

  2. Ostle, Turner, Hicks, and McElrath (1996). Engineering Statistics: The Industrial Experience. Duxbury Press. ↩