Definition of X-linked in Biology.

When a gene being examined is present on the X chromosome, but not on the Y chromosome, it is said to be X-linked.
Eye color in Drosophila was one of the first X-linked traits to be identified, and Thomas Hunt Morgan mapped this trait to the X chromosome in 1910.
Males are said to be hemizygous, because they have only one allele for any X-linked characteristic.
When they inherit one recessive X-linked mutant allele and one dominant X-linked wild-type allele, they are carriers of the trait and are typically unaffected.
Eye color in Drosophila is an example of a X-linked trait

At that time, he already knew that X and Y have to do with gender.
He was able to conclude that the gene for eye color was on the X chromosome.
This trait was thus determined to be X-linked and was the first X-linked trait to be identified.
Males are said to be hemizygous, in that they have only one allele for any X-linked characteristic.
In Drosophila, the gene for eye color is located on the X chromosome.

A definite integral is the area of the region in the $xy$ -plane bound by the graph of $f$ , the $x $-axis, and the vertical lines$ x=a $and$ x=b$.
Given a function $f$ of a real variable x and an interval $[a, b]$ of the real line, the definite integral $\int_{a}^{b}f(x)dx$ is defined informally to be the area of the region in the $xy $-plane bound by the graph of$ f $, the$ x $-axis, and the vertical lines$ x = a $and$ x=b $, such that the area above the$ x $-axis adds to the total, and that the area below the$ x$-axis subtracts from the total.
For example, consider the curve $y = f(x)$ between 0 and x = 1 with $f(x) = \sqrt{x}.$
As for the actual calculation of integrals, the fundamental theorem of calculus, due to Newton and Leibniz, is the fundamental link between the operations of differentiating and integrating.
Applied to the square root curve, $f(x) = x^{1/2}$, the theorem says to look at the antiderivative:

The fundamental theorem of calculus is a theorem that links the concept of the derivative of a function to the concept of the integral.
The fundamental theorem of calculus is a theorem that links the concept of the derivative of a function to the concept of the integral.
That is, $f$ and $F$ are functions such that, for all $x $in$ [a,b] $,$ F'(x) = f(x)$.
By definition, the derivative of $A(x) $is equal to$ \frac{A(x+h)−A(x)}{h} $as$ h$ tends to zero.
By replacing the numerator, $A(x+h)−A(x) $, by$ hf(x) $and dividing by$ h $,$ f(x)$ is obtained.

This area is represented by the probability P ( X < x ) .
The area to the right is then P ( X > x ) = 1 − P ( X < x ) .
Remember, P ( X < x ) = Area to the left of the vertical line through x.
P ( X > x ) = 1 − P ( X < x ) = .
P ( X < x ) is the same as P ( X ≤ x ) and P ( X > x ) is the same as P ( X ≥ x ) for continuous distributions.

for $\left | x \right | \leq 1$ (unless $x = -1$).
Substituting $x − 1 $for$ x $, we obtain an alternative form for$ \ln(x)$ itself:
$\ln(x) = (x - 1) - \dfrac{(x - 1)^{2}}{2} + \dfrac{(x - 1)^{3}}{3} - \cdots$
for $\left | x -1 \right | \leq 1$ (unless $x = 0$).
Here is an example in the case of $g(x) = \tan(x)$:

For $0 < x < \frac{ \pi}{2} $,$ \sin x < x < \tan x.$
$\displaystyle{\lim_{x \to 0} \left ( \frac{x}{\sin x} \right ) = 1}$
$\displaystyle{\lim_{x \to 0} \left ( \frac{\sin x}{x} \right ) = 1}$
$\displaystyle{\frac{(1−\cos x)(1+\cos x)}{x(1+\cos x)}=\frac{(1−\cos^2x)}{x(1+\cos x)}=\frac{\sin^2x}{x(1+\cos x)}= \frac{\sin x}{x} \cdot \frac{\sin x}{1+\cos x}}$
$\displaystyle{\lim_{x \to 0}\left ( \frac{\sin x}{x} \frac{\sin x}{1 + \cos x} \right ) = \left (\lim_{x \to 0} \frac{\sin x}{x} \right ) \left ( \lim_{x \to 0} \frac{\sin x}{1 + \cos x} \right ) = \left (1 \right )\left (\frac{0}{2} \right )= 0}$

This is sometimes called the unique effect of $x$ on $y$.
In contrast, the marginal effect of $x $on$ y $can be assessed using a correlation coefficient or simple linear regression model relating$ x $to$ y $; this effect is the total derivative of$ y $with respect to$ x$.
This may imply that some other covariate captures all the information in $x $, so that once that variable is in the model, there is no contribution of$ x $to the variation in$ y$.
In this case, including the other variables in the model reduces the part of the variability of $y$ that is unrelated to $x $, thereby strengthening the apparent relationship with$ x$.
In some cases, it can literally be interpreted as the causal effect of an intervention that is linked to the value of a predictor variable.

( x − $\bar{x}$ ) or ( x − µ ) = Deviations from the mean (how far a value is from the mean)
( x − $\bar{x}$)2 or ( x − µ )2 = Deviations squared
$\bar{x} = \frac{\sum{x}}{n} or x = \frac{\sum{f} \cdot x}{n}\bar{x} = \frac{\sum{x}}{n} or x = \frac{\sum{f} \cdot x}{n}$
$s = \sqrt{\frac{\sum(x \bar{x})^2}{n 1}}or s = \sqrt{\frac{\sum{f} \cdot ( x \bar{x})^2}{n-1}}$
$s = \sqrt{\frac{\sum(x \bar{x})^2}{N}}or s = \sqrt{\frac{\sum{f} \cdot ( x \bar{x})^2}{N}}$

$\displaystyle{\int u(x) v'(x) \, dx = u(x) v(x) - \int u'(x) v(x) \ dx}$
$dv = \cos(x)\,dx \\ \therefore v = \int\cos(x)\,dx = \sin x$
$\begin{aligned} \int x\cos (x) \,dx & = \int u \, dv \\ & = uv - \int v \, du \\ & = x\sin (x) - \int \sin (x) \,dx \\ & = x\sin (x) + \cos (x) + C \end{aligned}$
Similarly, the area of the red region is $A_2=\int_{x_1}^{x_2}y(x)dx$.
The total area, $A_1+A_2$ , is equal to the area of the bigger rectangle, $x_2y_2 $, minus the area of the smaller one,$ x_1y_1 $:$ \int_{y_1}^{y_2}x(y)dy+\int_{x_1}^{x_2}y(x)dx=\biggl.x_iy_i\biggl|_{i=1}^{i=2}$.

X-linked