Abstract
In many applications, the outcomes of a probabilistic experiment are numbers or have some numbers associated with them, and we can use these numbers to obtain important information beyond what we have seen so far.
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Notes
- 1.
Actually, in infinite sample spaces, there exist complicated functions for which not all such sets are events, and so we define a r.v. as not just any real-valued function X, but a so-called measurable function, that is, one for which all such sets are events. We shall ignore this issue; it is explored in more advanced books.
- 2.
Sometimes f(x) is considered to be a function on all of \(\mathbb{R}\) , with f(x) = 0 if x is not a possible value of X. This is a minor distinction, and it should be clear from the context which definition is meant.
- 3.
Note that we are using the same letter f for this function as for the p.f. of a discrete r.v. This notation cannot lead to confusion though, since here we are dealing with continuous random variables rather than discrete ones. On the other hand, using the same letter for both functions will enable us to combine the two cases in some formulas later.
- 4.
The function f is not unique, because the integral remains unchanged if we change the integrand in a countable number of points. Usually, however, there is a version of f that is continuous wherever possible, and we shall call this version the density function of X, ignoring the possible ambiguity at points of discontinuity.
- 5.
The symbol \(\sim \) means that the ratio of the expressions on either side of it tends to 1 as dx tends to 0 or, equivalently, that the limits of each side divided by dx are equal.
- 6.
We frequently use the words “point” and “number” interchangeably, ignoring the distinction between a number and its representation on the number line, just as the word “interval” is commonly used for both numbers and points.
- 7.
f is not unique: its values can be changed at a countable number of points, such as a and b, for instance, without affecting the probabilities, which are integrals of f.
- 8.
P(X = x, Y = y) stands for P(X = x and Y = y) = P \((\left \{X = x\right \} \cap \left \{Y = y\right \}).\)
- 9.
Sometimes f(x,y) is defined for all real numbers x,y, with f(x,y) = 0 if P(X = x) = 0 or P(Y = y) = 0.
- 10.
The adjective “marginal” is really unnecessary; we just use it occasionally to emphasize the relation to the joint distribution.
- 11.
\(F(x,\infty )\) is a shorthand for lim \(_{y\rightarrow \infty }F(x,y)\) , etc.
- 12.
The same ambiguities arise as in the one-dimensional case. (See footnote \(^{\mbox{ 5.2.1}}\) on page 5.2.1.)
- 13.
That is, a Cartesian product of n intervals, one from each coordinate axis.
- 14.
Note that it makes no difference for this assignment of probabilities whether we consider the region D open or closed or, more generally, whether we include or omit any set of points of dimension less than n.
- 15.
More precisely, A is any set in \(\mathbb{R}^{2}\) such that {s: (X(s), Y (s)) ∈ A} is an event.
- 16.
- 17.
Note that the max and the min have to be taken pointwise, that is, for each sample point s, we have to consider the max and the min of \(\{X_{1}\left (s\right ),X_{2}\left (s\right ),\ldots,X_{n}\left (s\right )\},\) and so Y and Z will in general be different from each of the X i.
- 18.
Recall that the symbol \(\sim \) means that the ratio of the expressions on either side of it tends to 1 as dx and dz tend to 0.
- 19.
We assume 00 = 1 where necessary.
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Schay, G. (2016). Random Variables. In: Introduction to Probability with Statistical Applications. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-30620-9_5
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DOI: https://doi.org/10.1007/978-3-319-30620-9_5
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