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NotesSheet.tex
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\begin{document}
\begin{center}
STAT 4033
\end{center}
Exam 1 Notes \hfill Name: \uline{Cooper Morris}
\begin{multicols}{2}
\textbf{\uline{C1S1:}}\\
\textbf{Population:} the entire set of all potential measurement\\
\textbf{Sample:} any subset of a population\\
\textbf{Simple Random Sample:} A sample of size \textit{n} taken in such a way that any group of size \textit{n} has the same chance of being selected\\
\textbf{Sampling Variability:} different samples from the same population can lead to differences\\
\textbf{Stratified Random Sampling:} the population is broken into groups based off a characteristic. Then a SRS is taken from each group\\
\textbf{Cluster Sampling:} target population has many groups, groups are selected by SRS of the groups. All elements of each group are selected\\
\textbf{Systematic Sample:} A listing is generated over time, every \textit{k}\textsuperscript{th} member is included in the sample.\\
\textbf{Tangible Population:} A population composed of members/individuals that exist.\\
\textbf{Conceptual Population:} A population composed of all values that can potentially be observed. They do not necessarily exist at any point in time.\\
\textbf{Observations:} The measurement, or set of measurements recorded from any individual in a sample.\\
\textbf{Variables:} The characteristics being observed from individuals.\\
\textbf{Quanitative Variables:} Possible values that represent \textit{quantiles of something.} Numbers of things.\\
\textbf{Ratio Variables:} Inherent zero value and ratios between values make sense.\\
\textbf{Interval Variables:} No meaningful ratios and arbitrary zero \\
\textbf{Qualitative:} A variable that takes a category of possible values.\\
\textbf{Nominal:} Ordering of categories makes sense.\\
\textbf{Ordinal:} No inherent ranking in categories.\\
\textbf{Observational Study:} Observe a sample from a population with minimal interaction.\\
\textbf{Experimental Study:} A study performed where the environment of subjects is strictly controlled.\\
\textbf{Response Variable(s):} The variable(s) of interest in a study.\\
\textbf{Explanatory Variable:} Variables to explain changes in the response variable.\\
\textbf{Confounding Variable(s):} Variables unaccounted for i a study that may explain changes in the response variable.\\
\textbf{\uline{C1S2:}}\\
\textbf{Measures of Central Tendency:} Values that represent where the``center" of a dataset is located.\\
\textbf{Measures of Variability:} Values that indicate how spread out the data are.\\
\textbf{Mode:} The measurement that occurs most often.\\
\textbf{Median:} The middle value in an ordered set.\\
\textbf{Mean:} The sum of all measurements divided by the total number of measurements.\\
\textbf{p\% Trimmed Mean:} The p\% lowest values and p\% of the highest values are removed from data, mean is taken.\\
\textbf{p\textsuperscript{th} percentile:} Value such that p\% of observations are at or below and (100-p)\% are above.\\
\textbf{Range:} difference between largest and smallest data points.\\
\textbf{Five Number Summary:} Min, Max, Median, Q1, and Q3\\
More \uline{relative} variation is higher CV, less \uline{relative} variation is lower CV.\\
\textbf{\uline{C1S3:}}\\
\textbf{Histogram:} Number of classes should be smallest whole number K that makes $2^K \geq$ number of measurements. For large data sets either $\log_2(n)$ or $2n^{1/3}$\\
\textbf{Unimodal:} One major peak\\
\textbf{Bimodal:} Two major peaks\\
\textbf{Symmetric:} Symmetric\\
\textbf{Right Skewed:} Long right tail, short left tail\\
\textbf{Boxplots:} Outliers are outside 1.5$\times$IQR. Box goes from $Q_1$ to $Q_3$, horizontal line at median, whiskers to largest data point inside 1.5$\times$IQR, X's for outliers\\
\end{multicols}
\newpage
\begin{multicols}{2}
\textbf{\uline{C2S1:}}\\
\textbf{Probability:} the chance that something happens.\\
\textbf{Experiment:} A process with an uncertain \uline{outcome}\\
\textbf{Sample Space ($\mathscr{S}$):} The set of all possible outcomes in an experiment.\\
\textbf{Outcome:} Each individual and non-reducible element of a sample space\\
\textbf{Event:} A set of 1 or more outcomes\\
\textbf{Union:} For events A and B the union is all outcomes in A, B, or Both. A$\cup$B\\
\textbf{Intersection:} The set of outcomes that are in both A and B. A$\cap$B\\
\textbf{Complement:} The set of outcomes in the sample space not in A. $A^C$\\
\textbf{Mutually Exclusive Events:} Events that share no outcomes in common.\\
\textbf{\uline{C2S3:}}\\
\textbf{Conditional Probability:} The probability of event A, given that event B has occurred P(A$\vert$B)\\
\textbf{Independent Events:} If two events don't affect each other\\
Two events are independent if and only if: P(A$\vert$B) = P(A)\\
\textbf{Exhaustive set of events:} A set of events is exhaustive of the sample space if their union is equivalent to the whole sample space.\\
\textbf{Bayes' Theorem:}\\
$P(A\vert B) = \frac{P(A\cap B)}{P(B)} = \frac{P(B\vert A)\cdot P(A)}{P(B)}$\\
$P(A\vert B) = \frac{P(B\vert A)\cdot P(A)}{P(B\vert A)\cdot P(A) + P(B\vert A^C)\cdot P(A^C)}$\\
$P(A_i\vert B) = \frac{P(B\vert A_i)\cdot(P(A_i)}{\sum_{j=1}^n P(B\vert A_j)\cdot P(A_j)}$
\vfill
\columnbreak
\vspace*{\fill}
Sample Mean: $\bar{y}$=$\frac{\sum_{i=1}^n y_i}{n}$\\
First Quartile: $Q_1=y_{25\%}$\\
Second Quartile: $Q_2=\textnormal{median}$\\
Third Quartile: $Q_3=y_{75\%}$\\
Interquartile Range: $IQR=Q_3-Q_1$\\
Sample Variance: $s^2=\frac{\sum_{i=1}^n (y_i-\bar{y})^2}{n-1}$\\
Sample Variance: $s^2=\frac{\sum_{i=1}^n y_i - \frac{(\sum y_i)^2}{n}}{n-1}$\\
Sample Standard Deviation: $s=\sqrt{s^2}$\\
Coefficient of Variation: $CV=\frac{\sigma}{\lvert\mu\rvert}$\\
Histogram Classes: $2^K \geq$ Measurements\\
Histogram Large Set Classes: $\log_2(n)$ or $2n^{1/3}$\\
Histogram Class Length: $\frac{Max-Min}{K}$\\
P($\mathscr{S}$) = 1\\
$0\leq P(A) \leq 1$\\
If A \& B are mutually exclusive:\\ P(A$\cup$B) = P(A) + P(B)\\
P(A\textsuperscript{C}) = 1 - P(A)\\
P(A$\cup$B) = P(A) + P(B) - P(A$\cap$B)\\
For \textit{k} operations with $n_i$ ways the sequence of \textit{k} operations:\\
\begin{displaymath}
\prod_{i=1}^k n_i
\end{displaymath}
Permutation of \textit{n} objects: n!\\
Permutation of \textit{k} objects from \textit{n} objects: $\frac{n!}{(n-k)!}$\\
Combinations of \textit{k} objects from \textit{n} objects: $\frac{n!}{k!(n-k)!}$\\
Combinations of groups with \textit{n} objects, \textit{r} groups, and group sizes $k_i$: $\frac{n!}{k_1!\cdot k_2! \cdot k_3! \ldots k_r!}$\\
\textit{N} total outcomes and $n_a$ outcomes for event A has probability: P(A) = $\frac{n_a}{N}$\\
P(A$\vert$B) = $\frac{P(A\cap B}{B}$\\
P(B$\vert$A) = $\frac{P(A\cap B}{A}$\\
P(A$\cap$B) = P(A$\vert$B)$\cdot$ P(B) = P(B$\vert$A)$\cdot$ P(A)\\
Independent Events:\\ P(A$\vert$B) = P(A) $\leftrightarrow$ P(B$\vert$A) = P(B)\\
Multiplication Rule for Independent Events:\\ P(A$\cap$B) = P(A)$\cdot$P(B)\\
$A_1,A_2,\ldots,A_N$ are a set of mutually exclusive events exhaustive of a sample space:\\
\begin{displaymath}
P(B) = \sum_{i=1}^n P(A_i\cap B)
\end{displaymath}
If $P(A_i) \neq 0$ for each $A_i$
\begin{displaymath}
P(B) = \sum_{i=1}^nP(B\vert A_i)\cdot P(A_i)
\end{displaymath}
\end{multicols}
\end{document}