What does a z-score of –1.5 mean? A.The mean is 1.5 standard deviations less than the measurement. B.There is an error in the measurement. C.The measurement is 1.5 standard deviations less than the mean. D.The variance cannot be calculated because the z-score is negative.

'Note that a sample of random variables (i.e. a set of measurable functions) must not be confused with the realizations of these variables (which are the values that these random variables take). In other words, $X_i$ is a function representing the measurement at the $i$-th experiment and $x_i=X_i(\omega )$ is the value we actually get when making the measurement.''
I'm afraid I don't understand this passage, can anybody please explain the point?

The equation $d=\frac{|Ax+By+Cz+D|}{(\sqrt{A^2+B^2+C^2})}$ gives is the distance between a plane and a point as the value 'd'. What is the unit of this value? Why is the distance between the plane and the point not a vector?

Let $\mu$ is finite measure and $\mu (A\mathrm{△}B)=0$.
Show that $\mu (A)=\mu (B)$
My work:
$\mu (A\mathrm{△}B)=0=\mu (A|B)+\mu (B|A)=(\mu (A)-\mu (A\cap B))+(\mu (B)-\mu (B\cap A))$
So $\mu (A)=-\mu (B)+2\mu (B\cap A)$
How is $\mu (A)=\mu (B)$?

I have a prediction $f(x)$ of some continuous process variable, based on an input variable $x$ (think: location). The prediction is incorrect, with the error being normal distributed with expected value $\mu$ and standard deviation $\sigma$.
Hence, the probability density function of $f(x)$ should be
$\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{((f(x)-\mu )/\sigma {)}^2}{2}}$
Is this correct? (No it is not, see answer below)
Now, I have a measurement $m$ of the process variable, based on an unknown input variable $x_m\cdot m$ is assumed to be correct, but quantized to integral numbers.
Given a set of $x_i$ together with their predictions $f(x_i)$, how can I compute a probability that $x_m$ was in the vicinity of $x_i$?
I apologize if the question makes no sense.

What are quantitative data measurements?
A. Measurements that are appropriate for any type of variable
B. Measurements of categorical variables and can be displayed as types or descriptions
C. Measurements that are appropriate in all experiments
D. Measurements of numerical variables and are displayed as numerical values

Two independent measurements are made of the lifetime of a charmed strange meson. Each measurement has a standard deviation of $7\times {10}^{-15}$ seconds. The lifetime of the meson is estimated by averaging the two measurements.
What is the standard deviation of this estimate ?

I need to give an example of a measure $\mu$ and subsets $A_n$ s.t.
$A_1\supset A_2\supset ...$
and
$\mu ({\cap }_{n=1}^{\mathrm{\infty }}{A}_n)\ne \underset{n\to \mathrm{\infty }}{lim}\mu (A_n)$
I hat the following in mind. Take $\mu$ to be the counting measure and $A_1=\mathbb{N},A_2=\mathbb{N}\setminus \{1\},...$ then
$\mu ({\cap }_{n=1}^{\mathrm{\infty }}{A}_n)=\mu (\mathrm{\varnothing })=0$
but now I have some strugles to show the other part with the limes. could someone help me please? It would be nice if we could procede with this example since this was my own idea without using any internet.