X-linked

Examples of X-linked in the following topics:

• Sex-Linked Traits

  • 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

• Chromosomal Theory of Inheritance

  • 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.

• The Definite Integral

  • 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

  • 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.

• Areas to the Left and Right of x

  • 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.

• The Natural Logarithmic Function: Differentiation and Integration

  • 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 $g(x) = \tan(x)$:

• Trigonometric Limits

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

• Making Inferences About the Slope

  • 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.

• Summary of Formulas

  • ( 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}$
  • $s = \sqrt{\frac{\sum(x \bar{x})^2}{n 1}}or s = \sqrt{\frac{\sum{f} \cdot ( x \bar{x})^2}{n-1}}$
  • $s = \sqrt{\frac{\sum(x \bar{x})^2}{N}}or s = \sqrt{\frac{\sum{f} \cdot ( x \bar{x})^2}{N}}$

• Integration By Parts

  • $\displaystyle{\int u(x) v'(x) \, dx = u(x) v(x) - \int u'(x) v(x) \ dx}$
  • $dv = \cos(x)\,dx \\ \therefore v = \int\cos(x)\,dx = \sin x$
  • $\begin{aligned} \int x\cos (x) \,dx & = \int u \, dv \\ & = uv - \int v \, du \\ & = x\sin (x) - \int \sin (x) \,dx \\ & = x\sin (x) + \cos (x) + C \end{aligned}$
  • 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, $x_1y_1$: $\int_{y_1}^{y_2}x(y)dy+\int_{x_1}^{x_2}y(x)dx=\biggl.x_iy_i\biggl|_{i=1}^{i=2}$.