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  /Title (economic-statistics-midterm-2-2.pdf)
  /Creator (Cheatography)
  /Author (lunarorbit)
  /Subject (Economic Statistics - Midterm 2.2 Cheat Sheet)
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    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Economic Statistics - Midterm 2.2 Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{lunarorbit} via \textcolor{DarkBackground}{\uline{cheatography.com/216484/cs/47304/}}}
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  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
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   Updated 6th November, 2025.\\
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\begin{multicols*}{4}

\begin{tabularx}{3.833cm}{X}
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\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Probability basics (definitions \& rules)}}  \tn
% Row 0
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\mymulticolumn{1}{x{3.833cm}}{Probability Basics} \tn 
% Row Count 1 (+ 1)
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\mymulticolumn{1}{x{3.833cm}}{{\bf{What probability actually means:}}  \newline • Probability is a number that tells you how likely something is to happen. \newline • It's always between 0 and 1: \newline ~ ▸ 0 = impossible \newline ~ ▸ 1 = guaranteed \newline ~ ▸ 0.5 = 50\% chance \newline You can think of probability as the {\bf{long-run frequency}} of something happening if you repeated it a bunch of times. \newline ~        Example: if you flip a fair coin 1,000 times, about half the flips will be heads → P(Heads)=0.5.}  \tn 
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\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Key probability symbols}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{• Event: any outcome or collection of outcomes you're interested in. \newline % Row Count 2 (+ 2)
~ Example: "Rolling an even number" on a die → that's the event A = \{2, 4, 6\}. \newline % Row Count 4 (+ 2)
• Sample space (S): all possible outcomes. \newline % Row Count 5 (+ 1)
~ Example: rolling a die → \newline % Row Count 6 (+ 1)
~ S = \{1, 2, 3, 4, 5, 6\}. \newline % Row Count 7 (+ 1)
• P(A): "The probability that event A happens." \newline % Row Count 9 (+ 2)
~ Example: P(Even)=3/6=0.5.% Row Count 10 (+ 1)
} \tn 
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\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Types of events}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{• Disjoint / mutually exclusive: can't happen at the same time. \newline % Row Count 2 (+ 2)
~ Example: "Rolling a 3" and "Rolling a 4." \newline % Row Count 4 (+ 2)
• Independent: one happening doesn't change the chance of the other. \newline % Row Count 6 (+ 2)
~ Example: Coin toss and rolling a die — they don't affect each other.% Row Count 8 (+ 2)
} \tn 
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\begin{tabularx}{3.833cm}{X}
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\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Example: "At least one"}}  \tn
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\mymulticolumn{1}{x{3.833cm}}{"If there's a 0.004 chance a test gives a false positive, what's the probability that at least one out of 200 tests is a false positive?"  \newline % Row Count 3 (+ 3)
This uses the {\bf{complement rule}} — it's easier to find the chance that none are false positives, then subtract from 1. \newline % Row Count 6 (+ 3)
\{\{ac\}\}P(At least one)=1−P(None) \newline % Row Count 7 (+ 1)
Each test has a 0.996 chance of being fine → \newline % Row Count 8 (+ 1)
P(None)=0.996\textasciicircum{}200 \newline % Row Count 9 (+ 1)
So P(At least one)=1−0.996\textasciicircum{}200% Row Count 10 (+ 1)
} \tn 
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\begin{tabularx}{3.833cm}{X}
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\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Random Variables (RVs)}}  \tn
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\mymulticolumn{1}{x{3.833cm}}{What they are} \tn 
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\mymulticolumn{1}{x{3.833cm}}{A random variable is just a number that represents the outcome of something random. \newline Example: \newline Toss a coin twice. \newline X = number of heads. \newline Possible X values: 0, 1, or 2. \newline We can describe all the possible values of X and how likely they are. That list is called a {\bf{probability distribution.}}}  \tn 
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\begin{tabularx}{3.833cm}{X}
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\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Probability distribution}}  \tn
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\mymulticolumn{1}{x{3.833cm}}{A table or formula showing: \newline % Row Count 1 (+ 1)
Every possible value of X \newline % Row Count 2 (+ 1)
The probability of each value \newline % Row Count 3 (+ 1)
Must satisfy: \newline % Row Count 4 (+ 1)
~ 1. All probabilities are between 0 and 1. \newline % Row Count 5 (+ 1)
~ 2. They add up to 1.% Row Count 6 (+ 1)
} \tn 
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\begin{tabularx}{3.833cm}{X}
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\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Mean (Expected Value)}}  \tn
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\mymulticolumn{1}{x{3.833cm}}{The {\bf{expected value (E{[}X{]})}} or {\bf{mean (μₓ)}} tells you the average outcome in the {\emph{long run.}} \newline % Row Count 2 (+ 2)
~ E{[}X{]}=∑(x×P(X=x)) \newline % Row Count 3 (+ 1)
{\emph{Example (from table above):}} E{[}X{]}=0(0.25)+1(0.50)+2(0.25)=1 → On average, 1 head per 2 flips.% Row Count 5 (+ 2)
} \tn 
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\begin{tabularx}{3.833cm}{X}
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\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Variance \& Standard Deviation}}  \tn
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\mymulticolumn{1}{x{3.833cm}}{Var(X)=∑(x−μ)$^{\textrm{2}}$ P(X=x) / SD(X)= √Var(X)  \newline % Row Count 1 (+ 1)
{\emph{Variance}} = average of squared distances from the mean.  \newline % Row Count 3 (+ 2)
{\emph{Standard deviation}} = average distance.  \newline % Row Count 4 (+ 1)
If SD is small → values are close to the mean.% Row Count 5 (+ 1)
} \tn 
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\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Useful shortcuts}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{• {\emph{E{[}aX+b{]}=aE{[}X{]}+b}} \newline % Row Count 1 (+ 1)
(Multiply and add constants outside the expectation.) \newline % Row Count 3 (+ 2)
• {\emph{Var(aX+b)=a$^{\textrm{2}}$ Var(X)}} \newline % Row Count 4 (+ 1)
(Adding doesn't affect spread, multiplying stretches it.) \newline % Row Count 6 (+ 2)
• {\emph{E{[}X+Y{]}=E{[}X{]}+E{[}Y{]}}} \newline % Row Count 7 (+ 1)
• If X and Y independent: {\emph{Var(}}X{\emph{+}}Y{\emph{)=Var(}}X{\emph{)+Var(}}Y{\emph{)}} \newline % Row Count 9 (+ 2)
• If correlated: {\emph{Var(}}X{\emph{+}}Y{\emph{)}} = {\emph{Var(}}X{\emph{)+Var(}}Y{\emph{)+2ρσ}}x{\emph{σ}}y \newline % Row Count 11 (+ 2)
(ρ = correlation between {\emph{X}} and {\emph{Y}})% Row Count 12 (+ 1)
} \tn 
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\begin{tabularx}{3.833cm}{X}
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\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Example: Lottery}}  \tn
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\mymulticolumn{1}{x{3.833cm}}{You buy a \$1 ticket that pays \$500 if you win (probability = 1/1000). If you lose, you get \$0.} \tn 
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\mymulticolumn{1}{x{3.833cm}}{E{[}X{]}=(499)(0.001)+(−1)(0.999)=−0.5 \newline You lose 50$\frac{3}{4}$ on average each play → expected loss.}  \tn 
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\begin{tabularx}{3.833cm}{X}
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\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Sampling \& Sampling Distributions}}  \tn
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\mymulticolumn{1}{x{3.833cm}}{{\bf{Population vs Sample}}} \tn 
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\mymulticolumn{1}{x{3.833cm}}{• {\bf{Population:}} the entire group you care about (all students in a school). \newline • {\bf{Sample:}} the smaller group you actually measure (30 students). \newline We use samples to {\bf{estimate}} the truth about populations.}  \tn 
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\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Types of samples}}  \tn
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\mymulticolumn{1}{x{3.833cm}}{• {\bf{Simple Random Sample (SRS):}} every individual has an equal chance of being chosen. \newline % Row Count 2 (+ 2)
• {\bf{Stratified sample:}} population divided into groups (strata) → random sample from each. \newline % Row Count 4 (+ 2)
• {\bf{Cluster / multistage:}} randomly pick clusters, then pick within them. \newline % Row Count 6 (+ 2)
⚠️ {\bf{Voluntary response or convenience samples are biased!}} (not random → results unreliable).% Row Count 9 (+ 3)
} \tn 
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\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Sampling distribution}}  \tn
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\mymulticolumn{1}{x{3.833cm}}{If you take {\bf{many random samples}} and compute a statistic (like a mean or proportion) for each sample, the {\bf{distribution of those statistics}} is the sampling distribution. \newline % Row Count 4 (+ 4)
We study its: \newline % Row Count 5 (+ 1)
• {\bf{Center:}} average (should equal population value if unbiased) \newline % Row Count 7 (+ 2)
• {\bf{Spread:}} how much sample results vary \newline % Row Count 8 (+ 1)
• {\bf{Shape:}} often becomes bell-shaped for large samples% Row Count 10 (+ 2)
} \tn 
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\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Unbiased estimator}}  \tn
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\mymulticolumn{1}{x{3.833cm}}{A statistic is {\bf{unbiased}} if its sampling distribution's center equals the true population parameter. \newline % Row Count 3 (+ 3)
{\emph{E{[}X̄{]}= μ}} and {\emph{E{[}p̂​{]}= p}} \newline % Row Count 4 (+ 1)
Unbiased means, "on average, it hits the right answer."% Row Count 6 (+ 2)
} \tn 
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\end{tabularx}
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\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{How sample size affects variability}}  \tn
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\mymulticolumn{1}{x{3.833cm}}{Bigger sample → smaller variability (less spread). \newline % Row Count 2 (+ 2)
{\emph{SD(X̄)=σ/√n}} \newline % Row Count 3 (+ 1)
As {\emph{n}} increases, denominator gets bigger → SD gets smaller.% Row Count 5 (+ 2)
} \tn 
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\begin{tabularx}{3.833cm}{X}
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\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Law of Large Numbers \& Central Limit Theorem}}  \tn
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\mymulticolumn{1}{x{3.833cm}}{{\bf{Law of Large Numbers}}} \tn 
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\mymulticolumn{1}{x{3.833cm}}{When you take more and more samples, the sample mean X̄ will get closer and closer to the population mean μ. \newline ~ Example: the average of 10 coin flips might not be 0.5, but the average of 10,000 flips almost definitely will be close to 0.5.}  \tn 
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\begin{tabularx}{3.833cm}{X}
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\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Central Limit Theorem (CLT)}}  \tn
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\mymulticolumn{1}{x{3.833cm}}{This is {\emph{super important}} for exams.} \tn 
% Row Count 1 (+ 1)
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\SetRowColor{LightBackground}
\mymulticolumn{1}{x{3.833cm}}{Even if the original population is {\bf{not Normal}}, when you take {\bf{a large enough sample}}, the {\bf{distribution of sample means}} will look {\bf{Normal (bell-shaped)}}. \newline ~ {\emph{X̄ ∼N(μ, σ/√n)}} \newline Meaning: \newline Centered at μ (the true mean) \newline Spread = {\emph{σ/√n}}}  \tn 
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\begin{tabularx}{3.833cm}{X}
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\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Why it matters}}  \tn
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\mymulticolumn{1}{x{3.833cm}}{CLT lets you use {\bf{z-scores}} and Normal tables to find probabilities for sample means. \newline % Row Count 2 (+ 2)
*z = sample mean - μ / σ/√n) \newline % Row Count 3 (+ 1)
Then use the z-table (or calculator) to find probabilities.% Row Count 5 (+ 2)
} \tn 
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\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Example (CLT in action)}}  \tn
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\mymulticolumn{1}{x{3.833cm}}{A machine fills cereal boxes. \newline % Row Count 1 (+ 1)
μ = 500g, σ = 10g. \newline % Row Count 2 (+ 1)
Sample 25 boxes. \newline % Row Count 3 (+ 1)
{\emph{SD(X̄)}} = 10/√25 = 2 \newline % Row Count 4 (+ 1)
{\emph{P(X̄ \textless{} 496)}} = {\emph{P}} ({\emph{Z}} \textless{} 496-500/2) = {\emph{P}} ({\emph{Z}} \textless{} -2) = 0.0228 \newline % Row Count 6 (+ 2)
→ 2.28\% chance the sample average weight is below 496g.% Row Count 8 (+ 2)
} \tn 
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% That's all folks
\end{multicols*}

\end{document}