Z-transform

Examples of Z-transform in the following topics:

• Correlation

  • The solution lies with Fisher's z' transformation described in the section on the sampling distribution of Pearson's r.
  • The conversion of r to z' can be done using a calculator.
  • This calculator shows that the z' associated with an r of -0.654 is -0.78.
  • The Z for a 95% confidence interval (Z.95) is 1.96, as can be found using the normal distribution calculator (setting the shaded area to .95 and clicking on the "Between" button).
  • The r associated with a z' of -1.13 is -0.81 and the r associated with a z' of -0.43 is -0.40.

• The Standard Normal Distribution

  • The standard normal distribution is a normal distribution of standardized values called z-scores.
  • A z-score is measured in units of the standard deviation.
  • x = µ + ( z ) σ = 5 + ( 3 )( 2 ) = 11 (6.1)
  • The transformation z = (x − µ)/σ produces the distribution Z ∼ N ( 0,1 ) .

• Standard Normal Distribution

  • The first column titled "Z" contains values of the standard normal distribution; the second column contains the area below Z.
  • The area below Z is 0.0062.
  • There is no need to transform to Z if you use the applet as shown in Figure 2.
  • If all the values in a distribution are transformed to Z scores, then the distribution will have a mean of 0 and a standard deviation of 1.
  • This process of transforming a distribution to one with a mean of 0 and a standard deviation of 1 is called standardizing the distribution.

• Triple Integrals in Cylindrical Coordinates

  • In R3 the integration on domains with a circular base can be made by the passage in cylindrical coordinates; the transformation of the function is made by the following relation:
  • The domain transformation can be graphically attained, because only the shape of the base varies, while the height follows the shape of the starting region.
  • because the z component is unvaried during the transformation, the $dx\, dy\, dz$ differentials vary as in the passage in polar coordinates: therefore, they become: $\rho \, d\rho \,d\varphi \,dz$.
  • This method is convenient in case of cylindrical or conical domains or in regions where it is easy to individuate the $z$ interval and even transform the circular base and the function.
  • $f(\rho \cos \varphi, \rho \sin \varphi, z) = \rho^2 + z$

• Lorentz Transformations

• Sampling Distribution of Pearson's r

  • The variable is called z' and the formula for the transformation is given below.
  • The details of the formula are not important here since normally you will use either a table or calculator to do the transformation.
  • What is important is that z' is normally distributed and has a standard error of
  • The first step is to convert both 0.60 and 0.75 to their z' values, which are 0.693 and 0.973, respectively.
  • The standard error of z' for N = 12 is 0.333.

• Change of Variables

  • $\displaystyle {\iiint_D f(x,y,z) \, dx\, dy\, dz = \iiint_T f(\rho \cos \phi, \rho \sin \phi, z) \rho \, d\rho\, d\phi\, dz}$
  • $f(x,y,z) \longrightarrow f(\rho \cos \theta \sin \phi, \rho \sin \theta \sin \phi, \rho \cos \phi)$
  • $\displaystyle {\iiint_D f(x,y,z) \, dx\, dy\, dz \\ = \iiint_T f(\rho \sin \phi \cos \theta, \rho \sin \phi \sin \theta, \rho \cos \phi) \rho^2 \sin \phi \, d\rho\, d\theta\, d\phi}$

• Z-Scores and Location in a Distribution

  • Thus, a positive $z$-score represents an observation above the mean, while a negative $z$-score represents an observation below the mean.
  • $z$-scores are also called standard scores, $z$-values, normal scores or standardized variables.
  • The use of "$z$" is because the normal distribution is also known as the "$z$ distribution."
  • A raw score is an original datum, or observation, that has not been transformed.
  • This may include, for example, the original result obtained by a student on a test (i.e., the number of correctly answered items) as opposed to that score after transformation to a standard score or percentile rank.

• Tensors

  • We can work out how tensors transform by looking at a few examples.
  • Let's use the Lorentz matrix to transform to a new frame
  • Because transforms as a contravariant vector and doesn't transform, must transform as a covariant vector.
  • $\displaystyle \frac{d}{d\tau} \left [ \begin{array}{c} E \\ p_x \\ p_y \\ p_z \end{array} \right ] = \frac{\gamma q}{c} \left [ \begin{array}{cccc} 0 & E_x & E_y & E_z \\ E_x & 0 & B_z & -B_y \\ E_y & -B_z & 0 & B_x \\ E_z & B_y & -B_x & 0 \end{array} \right ] \left [ \begin{array}{c} c \\ v_x \\ v_y \\ v_z \end{array} \right ]$
  • This is called a duality transformation.

• The Standard Normal Curve

  • This type of random variable is often denoted by $Z$, instead of $X$.
  • Luckily, one can transform any normal distribution with a certain mean $\mu$ and standard deviation $\sigma$ into a standard normal distribution, by the $z$-score conversion formula:
  • The $z$-score gets its name because of the denomination of the standard normal distribution as the "$Z$" distribution.
  • $\displaystyle z=\frac { 70.4-64 }{ 2.5 } =\frac { 6.4 }{ 2.5 } =2.56$
  • Therefore, the probability $P(X>70.4)$ is equal to $P(Z>2.56)$, where $X$ is the normally distributed height with mean $\mu=64 \ \text{inches}$ and standard deviation $\sigma = 2.5 \ \text{inches}$ ($\{X \sim N(64, 2.5)\}$, for short), and $Z$ is a standard normal distribution $\{Z \sim N(0, 1)\}$.