OpenLLM-France/wikimedia huggingface dataset:
# Chess Opening Theory/1. e4/1...e5/2. Nf3/2...Nc6/3. Bb5/3...f5/4. Nc3/4...fxe4/5. Nxe4/5...d5/6. Nxe5/6...dxe4/7. Nxc6/7...Qd5/8. c4/8...Qd6/9. Nxa7/9...Bd7/10. Qh5/10...g6

| | a | b | c | d | e | f | g | h | |
| 8 | | | | | | | | | 8 |
| 7 | | | | | | | | | 7 |
| 6 | | | | | | | | | 6 |
| 5 | | | | | | | | | 5 |
| 4 | | | | | | | | | 4 |
| 3 | | | | | | | | | 3 |
| 2 | | | | | | | | | 2 |
| 1 | | | | | | | | | 1 |
| | a | b | c | d | e | f | g | h | |
11. Bxd7+ is necessary to free the b5-square if the knight is ever going to get out of a7.
# Chess Opening Theory/1. e4/1...e5/2. Nf3/2...Nc6/3. Bb5/3...f5/4. Nc3/4...fxe4/5. Nxe4/5...d5/6. Nxe5/6...dxe4/7. Nxc6/7...Qd5/8. c4/8...Qd6/9. Nxa7/9...Bd7/10. Qh5/10...g6/11. Bxd7

| | a | b | c | d | e | f | g | h | |
| 8 | | | | | | | | | 8 |
| 7 | | | | | | | | | 7 |
| 6 | | | | | | | | | 6 |
| 5 | | | | | | | | | 5 |
| 4 | | | | | | | | | 4 |
| 3 | | | | | | | | | 3 |
| 2 | | | | | | | | | 2 |
| 1 | | | | | | | | | 1 |
| | a | b | c | d | e | f | g | h | |
11...Qxd7 is necessary to avoid losing another pawn to 12.Qb5+. It appears to lose a rook instead by unprotecting e5, but accepting the gift turns out to be fraught with danger.
# Chess Opening Theory/1. e4/1...e5/2. Nf3/2...Nc6/3. Bb5/3...f5/4. Nc3/4...fxe4/5. Nxe4/5...d5/6. Nxe5/6...dxe4/7. Nxc6/7...Qd5/8. c4/8...Qd6/9. Nxa7/9...Bd7/10. Bxd7/10...Qxd7/11. Qh5

| | a | b | c | d | e | f | g | h | |
| 8 | | | | | | | | | 8 |
| 7 | | | | | | | | | 7 |
| 6 | | | | | | | | | 6 |
| 5 | | | | | | | | | 5 |
| 4 | | | | | | | | | 4 |
| 3 | | | | | | | | | 3 |
| 2 | | | | | | | | | 2 |
| 1 | | | | | | | | | 1 |
| | a | b | c | d | e | f | g | h | |
All of Black's moves in this position look terrible! 11...Qf7?? runs into 12.Qb5+. Moving the king out to e7 also loses at least a pawn to 12.Qe5+. The computer recommends 11...Kd8 as the least bad option, although White could then swing the queen over to a5 threatening Nc6+ and so forcing the king straight back to e8, while potentially saving a tempo by not having to move the knight immediately.
So the modern, active way of defending this position as Black is with 11...g6!? It doesn't sacrifice a whole rook, "only" an exchange, because White hasn't got time to capture on h8 and also save the knight.
# Chess Opening Theory/1. e4/1...e5/2. Nf3/2...Nc6/3. Bb5/3...f5/4. Nc3/4...fxe4/5. Nxe4/5...d5/6. Nxe5/6...dxe4/7. Nxc6/7...Qd5/8. c4/8...Qd6/9. Nxa7/9...Bd7/10. Bxd7/10...Qxd7/11. Qh5/11...g6

| | a | b | c | d | e | f | g | h | |
| 8 | | | | | | | | | 8 |
| 7 | | | | | | | | | 7 |
| 6 | | | | | | | | | 6 |
| 5 | | | | | | | | | 5 |
| 4 | | | | | | | | | 4 |
| 3 | | | | | | | | | 3 |
| 2 | | | | | | | | | 2 |
| 1 | | | | | | | | | 1 |
| | a | b | c | d | e | f | g | h | |
12. Qe5+ is necessary or Black will have a free move to capture on a7.
# Chess Opening Theory/1. e4/1...e5/2. Nf3/2...Nc6/3. Bb5/3...f5/4. Nc3/4...fxe4/5. Nxe4/5...d5/6. Nxe5/6...dxe4/7. Nxc6/7...Qd5/8. c4/8...Qd6/9. Nxa7/9...Bd7/10. Bxd7/10...Qxd7/11. Qh5/11...g6/12. Qe5/12...Kf7/13. Nb5/13...c6

| | a | b | c | d | e | f | g | h | |
| 8 | | | | | | | | | 8 |
| 7 | | | | | | | | | 7 |
| 6 | | | | | | | | | 6 |
| 5 | | | | | | | | | 5 |
| 4 | | | | | | | | | 4 |
| 3 | | | | | | | | | 3 |
| 2 | | | | | | | | | 2 |
| 1 | | | | | | | | | 1 |
| | a | b | c | d | e | f | g | h | |
It's now time for White to offer to take the queens off with 14. Qd4 and hope that the two extra pawns will be sufficiently healthy to win the day.
The retreat 14.Nc3 has been played too, but the attack on e4 is an illusion as long as the e-file is open for the a8-rook (14...Nf6 15.Nxe4? Nxe4 and now 16.Qxe4?? Re8 is a train wreck while 16.Qxh8 Nf6 is just a free piece for Black compared to the earlier Qxh8 lines) so the knight won't be well placed on c3.
# English-Hanzi/Week

A week is a period of 7 days.
一 星期 有 7天。
# English-Hanzi/Weight control can promote healthy blood sugar levels

Weight control can promote healthy blood sugar levels and has been shown to reduce the risk of type 2 diabetes.
控制体重可以促进健康的血糖水平，并且已经显示出可以减少2型糖尿病的风险。
# English-Hanzi/West Congo

West Congo refers to the Republic of the Congo. It is a country in the central of Africa. Its population is 5.2 million. Its capital is Brazzaville.
West Congo指Congo共和国。它是一个在非洲中部的国家。它的人口是520万。它的首都是Brazzaville。
# English-Hanzi/Weather

The weather is the condition of the atmosphere that is at a particular place and time.
天气 是 在 特定 地点 和 时间 的 大气 状况。
# English-Hanzi/We need more workers

As you know, we need more workers.
如你所知，我们需要更多的工人。
# English-Hanzi/We used to think that stretching muscles is necessary to warm the muscles and prepare for exercise

We used to think that stretching muscles is necessary to warm the muscles and prepare for exercise. However, more and more studies have shown that stretching muscles before warming up can actually damage muscles. When everything gets cold, the muscle fibers are not ready and may be damaged. If you exercise first, blood will flow to the area, which will make the tissue softer and easier to stretch.
我们曾经认为，伸展肌肉对于温暖肌肉和准备锻炼很必要。 但是，越来越多的研究表明，在热身之前伸展肌肉实际上会损害肌肉。 当一切变冷时，肌肉纤维尚未准备好，可能会受损。 如果您先运动，血液将流到该区域，这将使组织更柔软，更容易拉伸。
# English-Hanzi/Wednesday

Wednesday is the day after Tuesday.
星期三 是 星期二 之后 一日。
# English-Hanzi/We should physically active

Special:search/active prefix:English-Hanzi/
There is plenty of evidence that the body needs regular exercise. This is very important if you want to live healthily.
有 很多 证据 表明 身体 需要 定期 运动。如果 你 想 健康地 生活，这 是 非常 重要 的。
Medical proof that regular exercise can:
* Reduce coronary heart disease and stroke risk 35%
* Reduce the risk of type 2 diabetes 50%
* Reduce the risk of colon cancer 50%
* Reduce the risk of breast cancer 20%
* Reduce the risk of early death 30%
* Reduce the risk of osteoarthritis 83%
* Reduce the risk of hip fracture 68%
* Reduce the risk of falls 30% (in the elderly)
* Reduce the risk of depression 30%
* Reduce the risk of dementia 30%
医学证明，经常运动可：
* 降低冠心病和中风风险35％
* 降低2型糖尿病风险50%
* 降低结肠癌风险50％
* 降低乳癌风险20%
* 降低早死风险30％
* 降低骨关节炎风险83%
* 降低髋骨骨折风险68％
* 降低跌倒风险30%（在老年人中）
* 降低抑郁症风险30%
* 降低痴呆症风险30％
# English-Hanzi/Well-known feature of sleep is the dream

A well-known feature of sleep is the dream, an experience typically recounted in narrative form, which resembles waking life while in progress, but which usually can later be distinguished as fantasy.
睡眠的一个众所周知的特征是做梦，这种体验通常以叙事的形式叙述，类似于在进行醒着时的生活，但通常可以在以后被区分为梦幻。
# English-Hanzi/West

The west is the opposite direction of the east.
西方 是 东方 的 相反 方向。
# Econometric Theory/Prerequisite
### Linear Algebra

It's assumed that the reader understands basic linear algebra. Throughout this course it is expected that the reader has knowledge of matrix algebra as it will be important knowledge to follow what is called the matrix approach to the linear regression model. All that is covered in the link below.

## This is the previous content

(I want a book entirely written in matrix algebra so that it will be easier to write formulae and is shorter, despite the complexity. I will try to make some contributions to the book Linear Algebra - Whisky Brewer)
(Is it possible to put the Linear Algebra book at the final chapter of this book on econometric theory?)
It's assumed that the reader understands basic linear algebra. however it's of particular usefulness to know about the properties of a very common class of matrix called a Projection Matrix. These matrices are called projection matrices because they have a geometrical interpretation of projecting data from one plane to another. {Insert more on projection matrices} {Define P and Q} {Show the geometrical implication.}
| ← Statistical Packages | Econometric Theory |
# Econometric Theory/Proofs of properties of β1
## Linearity

To be linear, ${\hat {\beta }}_{1}$ must be a linear function of $Y_{i}$, as shown below
${\hat {\beta }}_{1}=\sum {k_{i}Y_{i}}$
where $k_{i}$ is a constant, at any given observation 'i'.

### Proof

From the deviation-from-means form of the solution of the OLS Normal Equation for ${\hat {\beta }}_{1}$, we have
${\hat {\beta }}_{1}={\frac {\sum {x_{i}y_{i}}}{\sum {x_{i}^{2}}}}={\frac {\sum {x_{i}(Y_{i}-{\bar {Y}})}}{\sum {x_{i}^{2}}}}={\frac {\sum {x_{i}Y_{i}}}{\sum {x_{i}^{2}}}}-{\frac {\sum {x_{i}{\bar {Y}}}}{\sum {x_{i}^{2}}}}$
$={\frac {\sum {x_{i}Y_{i}}}{\sum {x_{i}^{2}}}}$, since ${\sum {x_{i}}}=0$.
$=\sum {k_{i}Y_{i}}$, where $k_{i}={\frac {x_{i}}{\sum {x_{i}}}}$, which is a constant for any given 'i'-value.
# Econometric Theory/Regression versus Causation and Correlation
## Statistical and deterministic Relationships

A deterministic relationship implies that there is an exact mathematical relationship or dependence between variables. An example in physics is Newton's law of gravity: $F=k\left({\frac {m_{1}m_{2}}{r^{2}}}\right)$, where F, the force, is proportional to a constant, k, the mass of two objects, $m_{1}$ and $m_{2}$, and inversely to the square of the distance.
A random or stochastic relationship allows that there is some variation in a relationship. This is where probability distributions will enter later on. The relationship may not be exact due to
* measurement errors
* reporting errors
* computing errors
* other influence,
etc. One example is crop yield relative to rain fall. We may not be able to measure the amount of rain accurately (measurement error), we round off to one decimal point (reporting error), there is a bug in the computer software that computes the sum (computing error) and there might be other factors such the quality of fertilisers, the quality of the earth and pollution (other influences).

## Regression versus Causation

Regression deals with dependence amongst variables within a model. But it cannot always imply causation. For example, we stated above that rainfall affects crop yield and there is data that support this. However, this is a one-way relationship: crop yield cannot affect rainfall. It means there is no cause and effect reaction on regression if there is no causation.
In short, we conclude that a statistical relationship does not imply causation.

## Regression versus Correlation

Correlations form a branch of analysis called correlation analysis, in which the degree of linear association is measured between two variables. If we calculate the correlation between crop yield and rainfall, we might obtain an estimate of, say, 0.69. This is reasonably high to prove that there is a mathematical relationship between them.
There is a distinction in how we regard the relationship between rainfall and crop yield. In statistics, both variables are assumed to be variables with random error in them. Both are treated on an equal footing and there is no distinction between them..
In regression analysis, crop yield is the dependent variable and rainfall is the explanatory variable, according to our theory. The distinction is that the independent variable has no random component, all values are fixed from this distribution
This will be important in {section on measurement}.

## Notes

{references/}

## Wikipedia links

Correlation does not imply causation

## Bibliography

* Gujarati, D.N. (2003). Basic Econometrics, International Edition - 4th ed. McGraw-Hill Higher Education. pp. 22–24. ISBN 0-07-112342-3.
* ↑ Gujarati (2003, p. 23)
# Econometric Theory/Multiple Regression Analysis

Our first regressions (MLE and OLS) were bivariate. Our lines were simple, two variable lines. However, in most economic data, there are a multitude of possible independent things that can effect a dependent variable. So we can expand our explanatory functions to allow multiple independent variables.
Instead of our functions looking like $Y_{=}\alpha +\beta X_{i}+\epsilon _{i}$, our functions look like $Y_{i}=\beta _{0}+\beta _{1}X_{1,i}+\beta _{2}X_{2,i}+\cdots +\beta _{n}X_{n,i}+\epsilon _{i}$
By adding more variables and data to our model, we can hopefully get a better fit and understanding of the dependent variable. However, with the added variables come added problems that will misguide our model.

## Goodness of Fit

When we move to the multiple regression case, our goodness of fit looks much like it previously did in the bivariate case. TSS = ESS + RSS. Our R² = ESS/TSS) = 1 - (RSS/TSS). We can still use our Coefficient of Determination, R² (R² = ESS/TSS = 1 - (RSS/TSS)), but there is a problem associated with it. R² will never decrease because of an addition of a variable, whether or not it helps us explain our dependent variable. When we add a new variable to the function, the ESS is calculated over a larger set of variables, and ESS will be greater than or equal to what we had before. This will cause our R² to increase, even if the addition of our new variable hurts our model. There is a tool to fix this problem. R² is replaced with Adjusted R² which adjusts it for the added degrees of freedom. Adjusted R² is signified by adding a bar above the 'R.'
${\bar {R^{2}}}=1-({\frac {\hat {var(\epsilon )}}{\hat {var(Y)}}})$
# Econometric Theory/Multicollinearity

While running regressions on multiple explanatory variables, there often is the problem of two variables having the same effects on the dependent variable. For example, say that the reading level of your father and the reading level of your mother can predict your eventual max reading level (YourLevel = a + b*FatherLevel + c*MotherLevel + u). However, there probably is a high probability that your parents have similar reading levels, so that using both may be redundant. This is a problem known as Multicollinearity. The problem with this is bias in your estimators. Say we can find yours through your mother's level. YourLevel = a + c*MotherLevel + u. If we add on your father's level, we will increase the prediction for your level, although in real life your level will not increase at all.

## Detecting Multicollinearity

Detecting multicollinearity can be more difficult than in the above example. But the first step that should be taken is an examination of the theory that is being tested. Is it redundant to have both mother's level and father's level? If this does not yield any results, probably because the theory is more complex, causing multicollinearity to be hidden, several econometric techniques can be used to find problems.
1) Large changes in the estimated regression coefficients when a predictor variable is added or deleted. Running the regression first with 'FatherLevel' and then without it may yield large variation, indicating that there is an error.
2) Non-significant results of simple linear regressions. Obviously if we find that with both FatherLevel and MotherLevel that neither are significant, than again there is something strange happening signaling possible multicollinearity.
3) Estimated regression coefficients have an opposite sign from predicted If in a regression with both FatherLevel and MotherLevel, b is positive, but c is negative; we know from theory that a higher reading level of the mother does not cause the child to be a worse reader. This is a possible sign of multicollinearity.
4)formal detection-tolerance or the variation inflation factor (VIF)
    $\mathrm {tolerance} =1-R^{2},\quad \mathrm {VIF} ={\frac {1}{\mathrm {tolerance} }}.$
A tolerance of less than 0.1 indicates a multicollinearity problem.
# Econometric Theory/Normal Equations Proof

Below is the proof of the Normal Equations for OLS.
The goal of OLS is to minimize the sum of squared error terms to find the best fit, also called the Residual Sum of Squares (RSS). This is denoted by $\sum {\hat {\epsilon _{i}}}^{2}$.

# Defining the RSS

Known: ${\hat {\epsilon _{i}}}=Y_{i}-{\hat {Y_{i}}}=Y_{i}-\alpha -\beta X_{i}$
RSS = $\sum {\hat {\epsilon _{i}}}^{2}$ =
$=\sum (Y_{i}-{\hat {Y_{i}}})^{2}$
$=\sum (Y_{i}-{\hat {\alpha }}-{\hat {\beta }}X_{i})^{2}$

# Differentiate the RSS (so that we can then minimise it)

$min_{\alpha }\sum {\hat {\epsilon _{i}}}^{2}$ =
${\frac {\partial \sum {\hat {\epsilon _{i}}}^{2}}{\partial {\hat {\alpha }}}}=\sum 2{\hat {\epsilon _{i}}}{\frac {\partial {\hat {\epsilon _{i}}}}{\partial {\hat {\alpha }}}}=2\sum {\hat {\epsilon _{i}}}(-1)=2\sum (Y_{i}-{\hat {\alpha }}-{\hat {\beta }}X_{i})(-1)=0$
$min_{\beta }\sum {\hat {\epsilon _{i}}}^{2}$ =
${\frac {\partial \sum {\hat {\epsilon _{i}}}^{2}}{\partial {\hat {\beta }}}}=\sum 2{\hat {\epsilon _{i}}}{\frac {\partial {\hat {\epsilon _{i}}}}{\partial {\hat {\beta }}}}=2\sum {\hat {\epsilon _{i}}}(-X_{i})=2\sum (Y_{i}-{\hat {\alpha }}-{\hat {\beta }}X_{i})(-X_{i})=0$
So we have two equations:
$\sum (Y_{i}-{\hat {\alpha }}-{\hat {\beta }}X_{i})(-1)=0$
and
$\sum (Y_{i}-{\hat {\alpha }}-{\hat {\beta }}X_{i})(-X_{i})=0$ (The two(2) here is divided from both sides)
setting them both equal to $\sum Y_{i}$
We get
$\sum Y_{i}=n{\hat {\alpha }}+{\hat {\beta }}\sum X_{i}$ (This is the first OLS Normal Equation)
and
$\sum Y_{i}X_{i}={\hat {\alpha }}\sum X_{i}+{\hat {\beta }}\sum X_{i}^{2}$ (This is the second OLS Normal Equation)

# Solve the Normal Equations

Divide the first equation by n
${\frac {1}{n}}\sum Y_{i}={\hat {\alpha }}+{\frac {1}{n}}{\hat {\beta }}\sum X_{i}$
Leaves us with $\left(\sum W_{i}{\frac {1}{n}}={\bar {W}}\right)$
${\bar {Y}}={\hat {\alpha }}+{\hat {\beta }}{\bar {X}}\Leftrightarrow {\hat {\alpha }}={\bar {Y}}-{\hat {\beta }}{\bar {X}}$
Now we know how to get α(hat), we can work on β(hat)
$\sum Y_{i}X_{i}={\hat {\alpha }}\sum X_{i}+{\hat {\beta }}\sum X_{i}^{2}=[{\bar {Y}}-{\hat {\beta }}{\bar {X}}]\sum X_{i}+{\hat {\beta }}\sum X_{i}^{2}=[{\frac {(\sum X_{i})(\sum Y_{i})}{n}}]+{\hat {\beta }}[\sum X_{i}^{2}-{\frac {(\sum X_{i})^{2}}{n}}]$
We can move β(hat) to one side
${\hat {\beta }}={\frac {\sum Y_{i}X_{i}-{\frac {(\sum X_{i})(\sum Y_{i})}{n}}}{\sum X_{i}^{2}-{\frac {(\sum X_{i})^{2}}{n}}}}={\frac {\sum (x_{i}-{\bar {x}})(y_{i}-{\bar {y}})}{\sum (x_{i}-{\bar {x}})^{2}}}$
- n \bar{X} \bar{Y}
And now we have our Normal equations for OLS.
Since we have two equations and two unknowns, we are able to solve for them (${\hat {\alpha }},{\hat {\beta }}$).
# Econometric Theory/Probability Density Function (pdf)

The probability density function is a function that describe the probability of a continuous random variable. A continuous random variable is a variable that can have infinitely many values from –∞ to ∞, e.g. height, weight etc. The probability of a continuous normal random variable X can be described by a function f(x).
# Econometric Theory/Properties of OLS Estimators

OLS estimators have the following properties:
* Linear
* Unbiased
* Efficient: it has the minimum variance
* Consistent

# Linear

OLS estimators are linear functions of the values of Y (the dependent variable) which are linearly combined using weights that are a non-linear function of the values of X (the regressors or explanatory variables). So the OLS estimator is a "linear" estimator with respect to how it uses the values of the dependent variable only, and irrespective of how it uses the values of the regressors.

# Unbiased

Suppose that the population size is 100 for anything that we are studying. We use samples of size 10 to estimate the $\alpha$ and $\beta$ of the population. Everytime we use a different sample (a different set of 10 unique parts of the population), we will get a different $\alpha$ and $\beta$.
With the OLS method of getting $\alpha$ and $\beta$, we get a situation wherein after repeated attempts of trying out different samples of the same size, the mean (average) of all the $\alpha$ and $\beta$ from the samples will be equal to the actual $\alpha$ and $\beta$ of the population as a whole.
Basically, this means that if you do the exercise over and over again with different parts of the population, and then you find the mean for all the answers you get, you will have the correct answer (or you will be very close to it).
A biased estimator will yield a mean that is not the value of the true parameter of the population.

# Efficient: Minimum variance

This property is what makes the OLS method of estimating $\alpha$ and $\beta$ the best of all other methods. When there are more than one unbiased method of estimation to choose from, that estimator which has the lowest variance is best. (Variance is a measure of how far the different $\alpha$ and $\beta$ are from their mean; the variance is the average distance of an element from the average.)
An estimator (a function that we use to get estimates) that has a lower variance is one whose individual data points are those that are closer to the mean. This estimator is statistically more likely than others to provide accurate answers. The OLS estimator is one that has a minimum variance.
This property is simply a way to determine which estimator to use.
* An estimator that is unbiased but does not have the minimum variance is not good.
* An estimator that has the minimum variance but is biased is not good
* An estimator that is unbiased and has the minimum variance of all other estimators is the best (efficient).
The OLS estimator is an efficient estimator.

# Consistent

A consistent estimator is one which approaches the real value of the parameter in the population as the size of the sample, n, increases.
# Econometric Theory/Probability Density Function (PDF)
## Probability Mass Function of a Discrete Random Variable

A probability mass function f(x) (PMF) of X is a function that determines the probability in terms of the input variable x, which is a discrete random variable (rv).
A pmf has to satisfy the following properties:
* $f(x)={\begin{cases}P(X=x_{i})&{\mbox{for }}i=1,2,\cdots ,n\\0&{\mbox{for }}x\neq x_{i}\end{cases}}$
* The sum of PMF over all values of x is one:
    $\sum _{i}f(x_{i})=1.$

## Probability Density Function of a Continuous Random Variable

The continuous PDF requires that the input variable x is now a continuous rv. The following conditions must be satisfied:
* All values are greater than zero.
$f(x)\geq 0$
* The total area under the PDF is one
$\int _{-\infty }^{\infty }f(x)\,dx=1$
* The area under the interval [a, b] is the total probability within this range
$\int _{a}^{b}f(x)\,dx=P(a\leq x\leq b)$

## Joint Probability Density Functions

Joint pdfs are ones that are functions of two or more random variables. The function
${\begin{aligned}f(X\in A,Y\in B)&=\int _{A}\,\int _{B}f(x,y)\,dx\,dy\\&=0,{\mbox{if }}x\notin A{\mbox{ and }}y\notin B\\\end{aligned}}$
is the continuous joint probability density function. It gives the joint probability for x and y.
The function
${\begin{aligned}p(X\in A,Y\in B)&=\sum _{X\in A}\sum _{Y\in B}p(x,y)\\&=0,{\mbox{if }}x\notin A{\mbox{ and }}Y\notin y\\\end{aligned}}$
is similarly the discrete joint probability density function

## Marginal Probability Density Function

The marginal PDFs are derived from the joint PDFs. If the joint pdf is integrated over the distribution of the X variable, then one obtains the marginal PDF of y, $f(y)$. The continuous marginal probability distribution functions are:
$f(x)=\int _{y}^{B}f(x,y)dy$
$f(y)=\int _{x}^{A}f(x,y)dx$
and the discrete marginal probability distribution functions are
$p(x)=\sum _{y\in B}p(x,y)$
$p(y)=\sum _{x\in A}p(x,y)$

## Conditional Probability Density Function

$f(x\mid y)=P(X=x,Y=y)={\frac {f(x,y)}{f(y)}}$
$f(y\mid x)=P(Y=y,X=x)={\frac {f(x,y)}{f(x)}}$

## Statistical Independence

* Gujarati, D.N. (2003). Basic Econometrics, International Edition - 4th ed. McGraw-Hill Higher Education. pp. 870–877. ISBN 0-07-112342-3.
# Flora of New York/sect-link

{{{1}}}
# Flora of New York/sn
## Usage

yields:
* Ledum buxifolium
* Ledum buxifolium Bergius
* 1777. Ledum buxifolium Bergius
* 1888. G. nivalisssp. imperati Baker
* 1888. G. nivalisssp. imperati Baker
* 1950. A. psilostachyavar. coronopifolia
* 1950. Ambrosia psilostachyavar. coronopifolia
* auct. Salicornia mucronata non Lag.
* 2014. Sporobolus pumilusP.M.Peterson & Saarela
* non Sporobolus patens Swallen
# Flora of New York/spa1

Flora of New York/spa1/doc
# Flora of New York/ssp2

Flora of New York/ssp2/doc
# Flora of New York/sn/doc
## Usage

yields:
* Ledum buxifolium
* Ledum buxifolium Bergius
* 1777. Ledum buxifolium Bergius
* 1888. G. nivalisssp. imperati Baker
* 1888. G. nivalisssp. imperati Baker
* 1950. A. psilostachyavar. coronopifolia
* 1950. Ambrosia psilostachyavar. coronopifolia
* auct. Salicornia mucronata non Lag.
* 2014. Sporobolus pumilusP.M.Peterson & Saarela
* non Sporobolus patens Swallen