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|1rt 1.3 @Scale {Stat 231 Notes}
@SB |1rt {Andrei Thorp}
@SB |1rt {Spring 2011}

@BB
@BB Population @Defn {
    Target @Defn {theoretical best case population}
    @SB Study @Defn {actual population used for study}
    @SB Sample @Defn {subset of the study population, used for sampling}
    @SB Unit @Defn {individual atom of the population}
}

@BB PPDAC @Defn {Problem, plan, data, analysis, conclusion}

@BB {PPDAC -- Data} @Defn {Can be discrete (individualized), continuous (like real
numbers), categorical (not numerically valuable, like names), binary (categorical
for two categories).}

@BB {Data Transforms} @Defn {
    Monotone @Defn {after the data is transformed, the original monotonic order
is preserved.}
    @SB {Monotone Decreasing} @Defn {like monotone, but order is reversed after
the transform}
    @SB {Affine} @Defn {in linear form @M{y = Ax + B}}
    @SB {Log} @Defn {transform of @M{x --> log(x)}, where log is ln}
}

@BB Variate @Defn {characteristics of units}

@BB Attributes @Defn {functions on units}

@BB {Problem Types} @Defn {
    Discriptive @Defn {involves learning about some attribute}
    @SB Causative @Defn {finding a causal link between variates. One variate is
`explanatory' and the other is `response'}
    @SB Predictive @Defn {predict value of a response variate for a given unit}
}

@BB Mean @Defn {@M{alpha(cdot)}, the attribute defining the average of the
population}

@BB Error @Defn {through the study, error may occur:
    @SB {study error} @Defn @M{alpha(P sub {study}) - alpha(P sub {target})}
    @SB {sample error} @Defn @M{alpha(S) - alpha(P sub {study})}
}

@BB {PPDAC -- Plan} @Defn {
    Experimental @Defn {You can manipulate a study's variables.}
    @SB Observational @Defn {Studies which are observational only.}
}

@BB {Kernel Density Estimate} @Defn { a smoothing algorithm for datasets}

@BB Histogram @Defn {Continuous data is split into bins, and these bins are then
plotted as a bar graph.}

@BB {Symmetry"/"Skew} @Defn {Data plots may have a `tail' that goes left or right,
and this is the direction of the skew. If there is no skew, then the data is
symmetric}

@BB {Frequency Polygon} @Defn {Join the top centres of each histogram bar with
lines to get the polygon}

@BB {Cumulative Frequency Plot} @Defn {Each bin of a histogram includes all the
preceding bars}

@BB {Empirical Rule} @Defn {For symmetric, unimodal distributions, if you know
@M{mu} and @M{sigma}, then:
@BulletList
@LI {Approx @M{68%} of observations lie in @M{mu +- sigma}}
@LI {Approx @M{95%} of observations lie in @M{mu +- 2 sigma}}
@LI {Approx @M{99.73%} of observations lie in @M{mu +- 3 sigma}}
@EndList
}

@BB {Event Independence} @Defn @M{P(A "|" B) = P(A "|" not B) = P(A)}

@BB {Correlation} @Defn {@Math{r = {S sub XY} over sqrt {S sub XX S sub YY}}
where @SB @Math{S sub XX = sum from {i=1} to {n} (x sub i - x overbar) sup 2} and
@Math {S sub XY = sum from {i=1} to {n} ((x sub i - x overbar)(y sub i - y overbar))}
@SB The closer the correlation is to @M{+- 1}, the better the correlation.}

@BB {Time Series Plot} @Defn {A scatter plot that tracks against time: @M{(t, y(t)).}}

@BB grey @Colour @Box {Midterm 1 Notes} @Defn {@RawBulletList
@LI {Remember some probability stuff like union, conditional, independence}
@LI {Very little calculation}
@LI {Memorize some of the distributions so far}
@LI {Write"/"memorize distributons' likelihood formulae}
@EndList}

@BB {MLE of Poisson Distribution} @Defn {A random variable of the Poisson distribution
is @Math{P(Y=y)=f sub y (y) = {e sup {-lambda} lambda sup y} over y!}, for
M{y = 0, 1, 2, ...}
@SB So @Math{L(y sub 1, y sub 2, ..., y sub n; lambda) =
{e sup {-lambda} lambda sup {y sub 1}} over {y sub 1 !} cdot
{e sup {-lambda} lambda sup {y sub 2}} over {y sub 2 !} cdot ... cdot
{e sup {-lambda} lambda sup {y sub n}} over {y sub n !} =
{e sup {-n lambda} lambda sup {sum from {i=1} to {n} y sub i}} over
{prod from {i=1} to {n} y sub i !}
}
@SB Now, lets take the ln of everything to make it simpler.
@SB @Math{l(lambda) = -n lambda
+ln(lambda) sum from {i=1} to {n} y sub i - prod from {i=1} to {n} y sub i !}.
@SB Take the derivitive of this and maximize... @M{lambda hat = y overbar}
}

@BB {Quick Review of Partial Derivitives} @Defn {Consider the function @Math{f(x, y) =
3x sup 2 + 4xy sup 2 + 2y sup 2}. Find the partial derivatives with respect to:
@BulletList
@LI {x: @Math{{2f(x,y)} over 2x = 6x + 4y sup 2}}
@LI {y: @Math{{2f(x,y)} over 2y = 8xy + 4y}}
@EndList
Pretty much: just ignore the other variable, treat it as a constant.}

@BB {Simple Linear Regression MLEs} @Defn {
     @Math{alpha hat = y overbar - beta hat x overbar}
@SB  @Math{beta hat = {S sub {XY}} over {S sub XX}}
@SB  @Math{sigma hat sup 2 = 1 over n sum from {i=1} to {n} (y sub i -
alpha hat - beta hat x sub i ) sup 2 = 1 over n (S sub {YY} - beta hat S sub XY )}
}

@BB {Practice Problem for MLEs"/"Regression} @Defn {Consider the following dataset:

@SB { @OneRow {{@Underline x
//0.5f 100
//0.5f 200
//0.5f 300
//0.5f 400
//0.5f 500
//0.5f 600
//0.5f 700
} |2f
{@Underline y
//0.5f 40
//0.5f 50
//0.5f 50
//0.5f 70
//0.5f 65
//0.5f 65
//0.5f 80
}} |3f
@OneRow {x = lbs of fertilizer per acre
@SB y = wheat yield (bushels per acre)
@SB Determine: |1f {
@M{y hat = alpha hat + beta hat x}
/0.5f @M{sigma hat sup 2}
/0.5f @M{p hat = r = {S sub XY} over {sqrt{S sub XX S sub YY}}}
/0.5f @M{r sup 2}
}
}
}
@SB Check using R -- make the two vectors, then use `summary(lm(x "~" y))'
}





@BB blue @Colour {Important Formulae} @Defn {
@Math{sum from {i=1} to {n} i = {n(n+1)} over 2}
@SB @Math{sum from {i=1} to {n} i sup 2 = {(n(n+1))} over 2 cdot {2n+1} over 3}
@SB {Event Conditional} @Defn @Math{P(A bar B) = {P(A intersection B)} over P(B)}
@SB {Event Independence} @Defn @M{P(A "|" B) = P(A "|" not B) = P(A)}
}

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