Notes, summaries, assignments, exams, and exercises for Mathematics for University

Fundamental Theorems in Probability

The Law of Large Numbers

The Law of Large Numbers is a theorem in probability that describes the average behavior of a succession of random variables as the total number of variables increases. The theorem provides sufficient hypotheses to affirm that the sample average converges to the average of the expectations of the random variables involved.

Chebyshev's Theorem

Chebyshev's Theorem provides an upper bound on the probability that values fall outside a certain range from the average.

Bernoulli's Theorem

Bernoulli's Theorem is a special case of the Law of Large Numbers, which specifies that the approximate frequency of an occurrence converges to the probability p of that occurrence as the experiment is repeated....

Linear Algebra and Matrix Properties

1. State one property of a square matrix that will guarantee it is singular.
   There are many possibilities, some of which are: having determinant 0, having dependent columns, having dependent rows, and having a non-trivial null-space.
2. Can a system of linear equations have exactly 2 solutions?
   No, a system of linear equations can have either 0, 1, or infinitely many solutions.
3. What advantages does the Cholesky factorization have over Gaussian elimination?
   The Cholesky factorization only requires about half the operations of Gaussian elimination, and pivoting is never required for stability.
4. Can an n × n positive-definite matrix be singular? What about an n × n symmetric positive-definite matrix?
   The condition

Main Accounts Classification with Spanish Names

Spanish

English

Assets:

Current assets:

Cash

Bank accounts

Temporary investments

Accounts receivable (Clients)

Accounts receivable (Other debtors)

Notes receivable (< 1 year)

Supplies

Merchandise inventory

Prepaid rent (< 1 year)

Prepaid insurance (< 1 year)

Advance payment to suppliers (< 1 year)

Fixed assets:

Land

Buildings

Accumulated depreciation on buildings

Machinery and equipment

Accumulated depreciation on machinery and equipment

Transport equipment (or: trucks)

Accumulated depreciation on transport equipment

Communication equipment

Accumulated depreciation on communication equipment

Computer equipment

Accumulated depreciation on computer equipment

Office furniture and equipment

Accumulated depreciation...

Quantitative Variables

1. Discrete: values of x have "gaps" between them (ex: test scores)

2. Continuous: no gaps between x values

Random Variable X

Assigns a number to outcomes of an experiment

Probability Distribution of a Discrete X

A table, graph, or forumla that shows the values of X and P(X) where:

Expected Value of Discrete X

1. The probability distribution of x represents a population (μ,σ^2,σ)

2. Mean of X = expected value of X

(1) Discrete in General

Variance of Discrete X

Standard Deviation of Discrete X

Ex:

(2) Binomial X

• Need n, P and q
• x = # of successes in n
• P(X) use binomial tables (if n < 5) or="" the="">

• mean --> μ = nP   variance --> σ^2 = nPq   standard deviation --> σ = √nPq
• NOTE: Formula gives

Measures of Position

Central Position Measures: The Arithmetic Mean

Central position measures: they describe the global behavior of the data distribution and locate the center of frequencies, that is, the place where the parts of the population are located.

Arithmetic mean: it is the sum of the values of the data multiplied by their relative frequencies. It is sensitive to extreme values.

Properties:

1. The sum of the deviations from the mean is zero.
2. Grouping.
3. Translation and change of scale.

Geometric Mean and Harmonic Mean

Geometric mean: it is the product of all the values raised to the power of the total number of values (the nth root). The values must be positive. It is used for percentages, rates, and index numbers.

Harmonic mean: it is used for...

Experimental Procedure

Period Depending on the Mass of Oscillation

Two different masses were chosen for the experiment: one metal and one wood. Measurements were taken twice for 10 swings per mass, keeping the swing angle and the length of the pendulum equal. The procedure was repeated with the second mass. We determined the average time for each mass and calculated the period (T) using the equation: T = time / number of swings.

• The corresponding absolute error for each body was calculated through the formula: ΔT = Δt / n.
• The oscillation angle used was α = (10 ± 1)°.
• The pendulum length used was L = (31.5 ± 0.1) cm.
• An m vs. T plot was constructed.
• The experimental data are recorded in Table 1.

Period Depending on the Angle of Oscillation

• Three

--------------------------------   

  ----------------------------------------------------------

Write an interpretation of r^2 using the template in the Activity 2.1 Readings. We will do this one as a class.

Template: The proportion of the variation in the Y variable that is explained by the SLR model with the X variable is r^2.

For slope : Template: As x var increases by 1 unit, we predict y var  will increase/dec  by ____  y var units.

For y-intercept: When x var = 0 units, we predict that the y var  will be ____ units..

For SLR: Error = epsilon = y - yhat = y - (betahat0 + betahat1x)

SSE = residual1^2 + res. 2^2 +…+ res.  n^2

Standard error of regression = Root MSE (in SAS language)

The text lists six conditions for simple linear...

Priprema podataka

Priprema podataka: medijana <- median(data$price, na.rm = TRUE)
data$price[is.na(data$price)] <- medijana
data <- data[complete.cases(data$price), ]  izbacivanje NA vrednosti

Metrike evaluacije

#accuracy - procenat tačnih predikcija od ukupnog broja opservacija
#precision - udeo onih koje smo predvideli kao pozitivne, a koje su stvarno pozitivne
#recall - udeo stvarno pozitivnih koje smo predvideli kao pozitivne
#F1 - služi za evaluaciju modela kada su precision i recall u balansu

1. Klasifikaciona stabla

Izlazna varijabla mora biti faktor.

tree1.cm <- table(true = test.data$izlazna, predicted = tree1.pred)
tree1.eval <- getEvalMetrics(tree1.cm)
getEvalMetrics <- function(cm){...}

Krosvalidacija: library(e1071) -...

Key Auditing Terms and Definitions

3.1 Audit: A systematic, independent, and documented process for obtaining audit evidence (3.3) and evaluating it objectively to determine the extent of compliance with the audit criteria (3.2).

3.2 Criteria: A set of audit policies, procedures, or requirements. NOTE: Audit criteria are used as a benchmark against which we compare the audit evidence (3.3).

3.3 Audit evidence: Records, statements of fact, or any other information that are relevant to the audit criteria (3.2) and are verifiable. NOTE: Audit evidence can be qualitative or quantitative.

3.4 Audit findings: The results of the evaluation of the audit evidence (3.3) compiled against the audit criteria (3.2). NOTE: Audit findings can indicate either conformity...

q linux prmita obtnr in pantaya, mdiantl ls, 1st list contnido dl dl dirctorio / root / examn,

where d ls nombrs arxiv notif qdn co +

ls-q-m / root / examn

linux list prmita q to2 ls dl dirctorio actuals subdirctorios

ls-l | grp ^ drwxr-x

linux spcificarl acr q ql c prmita sal sy sy x dfcto xal root.

xs-s / bin / cs

/ bin / bas
clar
if tst-f $ 1
tn
if tst-d $ 2
tn
ls $ 2 | grp-w $ 1
if tst $? -eq 0
tn
exo-e $ 1 exist enl arxiv dirctorio $ 2 \ n
els
cp $ 1 $ 2
fi
els
exo-and dirctorio not valid \ n
exit
fi
els
exo-and arxiv not valid \ n
exit
fi

+ corxal preferentially to

awk program, yamado listado_imc prmita ache and gnrarl list arxiv prsnas.Dat d prsnas with imc

xa valors 24.99l notif 18.5 and s considrado normal.L imc