Originally Posted by

**LinguisticM**
My sentence was indeed not precise.

X is the position of the particle and thetaX comes from the direction of motion of the particle. This is the angle between the projection of the momentum vector onto the Y=0 plane and the Z direction. If theta and phi are the polar and azimutal angles in the spherical coordinate system, respectively, then the thetaX is:

thetaX = tan(theta)cos(phi)

Including Y and thetaY, there are altogether four random variables in this problem.

Since the correlation between X and Y is not the main focus here, one can treat X and Y as uncorrelated and use only two bivariate distributions to describe the beam profile i.e. one distribution with X,Xtheta and the other with Y,Ytheta.

Position and angle may or may not be correlated depending on e.g. if beamline focusing quadrupoles are active upstream of the measurement plane, the amount of material that the beam has traversed thereby losing the angle-position dependence as a result of multiple Coulomb scattering process. X and Y position parameters may also be correlated, which results in a rotated elliptical shape of the beam spot, and this is usually not desired. In this case I can see how the 'projected' and 'conditional' variances would be different, but this is dependent of the covariance and contrary to what I can find in the literature (see below).

Let me try explaining with words. If this won't work then I'll make some sketches.

This is an excerpt that discusses the difference between the conditional and projected quantities:

<quote start>

1/(2*Pi*sqrt(d))*exp(-1/2*((a*thetaX^2-2*b*X*thetaX+c*X^2)/d)

Random variables x and thetaX are projected, or 'plane' variables. For example, x is the x coordinate of the random variable r. The variance of x, denoted a, is however different from what would be measured in a typical beam profile measurement. In a such a measurement, assuming measurement in x direction along the y = 0 line, one obtains a conditional probability distribution of x coordinate with the condition y = 0. This is not the same as the probability distribution for the x coordinate; they are both Gaussians, but the variance of the conditional is larger by a factor of 2. Sometimes such conditional variables are called 'space' variables

<quote end>

"d" above is the determinant of the covariance matrix.

<quote start>

At any x we can also calculate a scattering power

T= d<theta^2>/dx

the rate of increase with x of the mean squared projected multiple Coulomb scattering MCS angle. In the early literature, T is the rate of increase in the mean squared space angle, which is greater by a factor 2. In transport calculations projected quantities are usually more convenient.

<quote end>

The above is another example. This time it refers to the angle thetaX.

Does this shed some light?