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Probability textbooks tend to be too simple, ignoring many important concepts and succumbing to the pedagogical issues we have discussed, or focus on the myriad technical details of probability theory and hence quickly fall beyond the proficiency of many readers. My favorite treatment of the more formal details of probability theory, and its predecessor measure theory, is Folland (1999) who spends significant time discussing concepts between the technical details.

2.1 Probability Distributions

From an abstract perspective, probability is a positive, conserved quantity which we want to distribute across a space, X . We take the total amount of this conserved quantity to be 1 with arbitrary units, but the mathematical consequences are the same regardless of this scaling. From this perspective probability is simply any abstract conserved quantity – in particular it does not refer to anything inherently random or uncertain.

A defines a mathematically self-consistent allocation of this conserved quantity across X . Letting A be a sufficiently well-defined subset of X , we write P π [ A ] as the probability assigned to A by the probability distribution π . Importantly, we want this allocation to be self-consistent – the allocation to any collection of disjoint sets, A n A m = 0 , n m , should be the same as the allocation to the union of those sets, P π [ N n = 1 A n ] = N n = 1 P π [ A n ] . In other words, no matter how we decompose the space X , or any well-defined subsets of X , we conserve probability.

For a finite collection of sets this self-consistency property is known as and would be sufficient if there were only a finite number of well-defined subsets in X . If we want to distribute probability across spaces with an infinite number of subsets, such as the real numbers, however, then we need to go a bit further and require self-consistency over any countable collection of disjoint sets, P π [ n = 1 A n ] = n = 1 P π [ A n ] . In particular, this property allows us to cover complex neighborhoods, such as that enclosed by a smooth surface, with an infinite collection of sets and then calculate the probability allocated to that neighborhood.

In addition to self-consistency we have to ensure that we assign all of the total probability in our allocation. This requires that all of the probability is allocated to the full space, P π [ X ] = 1 .

These three conditions completely specify a valid probability distribution, although to be formal we have to be careful about what we mean by “well-defined” subsets of X . Somewhat unnervingly we cannot construct an object that self-consistently allocates probability to subset of X because of some very weird, pathological subsets. Fortunately the same properties that make these subsets pathological also prevent them from belonging to any σ -algebra, consequently we can construct our probability distribution relative to a given σ -algebra, X .

Formally, then, probability theory is defined by , which we can write as:

The more familiar rules of probability theory can all be derived from these axioms. For example the last self-consistency condition implies that P π [ A ] + P π [ A c ] = P π [ X ] = 1 or P π [ A ] = 1 P π [ A c ] .

A probability distribution is then completely specified by the ( X , X , π ) which is often denoted more compactly as x π where x X denotes the space, π denotes the probability distribution, and a valid σ -algebra is assumed.

2.2 Expectation Values

The allocation of probability across a space immediately defines a way to summarize how functions of the form f : X R behave. , E π [ f ] , reduce a function to a single real number by averaging the function output at every point, f ( x ) , weighted by the probability assigned around that point. This weighting process emphasizes how the function behaves in neighborhoods of high probability while diminishing its behavior in neighborhoods of low probability.

How exactly, however, do we formally construct these expectation values? The only expectation values that we can immediately calculate in closed form are the expectations of an that vanishes outside of a given set, I A [ x ] = { 1 , x A 0 , x A . The expectation of an indicator function is simply the weight assigned to A , which is just the probability allocated to that set, E π [ I A ] P π [ A ] . We can then build up the expectation value of an arbitrary function with a careful approximation in terms of these indicator functions in a process known as . For more detail see the following optional section.

When our space is a subset of the real line, X R , there is a natural of X into R , ι : X R x x . For example this embedding associates the natural numbers, { 0 , 1 , 2 , } , with the corresponding values in the real line, or the interval [ 0 , 1 ] with the corresponding interval in the full real line.

In this circumstance we define the of the probability distribution as m π = E π [ ι ] , which quantifies the location around which the probability distribution is focusing its allocation. Similarly we define the of the probability distribution as V π = E π [ ( ι m π ) 2 ] , which quantifies the breadth of the allocation around the mean. We will also refer to the variance of an arbitrary function as V π [ f ] = E π [ ( f E π [ f ] ) 2 ] .

While we can always define expectation values of a function f : X R , a probability distribution will not have a well-defined mean and variance unless there is some function whose expectation has a particular meaning. For example, if our space is a subset of the real numbers, X R N , then there is no natural function whose expectation value defines a scalar mean. We can, however, define means and variances as expectations of the , ˆ x n : R N R , that project a point x X onto each of the component axes. These component means and variances then provide some quantification of how the probability is allocated along each axis.

2.3 Extra Credit: Lebesgue Integration

As we saw in Section 2.2 only the indicator functions have immediate expectation values in terms of probabilities. In order to define expectation values of more general functions we have to build increasingly more complex functions out of these elementary ingredients.

The countable sum of indicator functions weighted by real numbers defines a , ϕ = n a n I A n . If we require that expectation is linear over this summation then the expectation value of any simple function is given by E π [ ϕ ] = E π [ n a n I A n ] = n a n E π [ I A n ] = n a n P π [ A n ] . Because of the countable additivity of π and the boundedness of probability, the expectation of a simple function will always be finite provided that each of the coefficients a n are themselves finite.

We can then use simple functions to approximate an everywhere-positive function, g : X R + . A simple function with only a few terms defined over only a few sets will yield a poor approximation to g , but as we consider more terms and more sets we can build an increasingly accurate approximation. In particular, because of countable additivity we can construct a simple function bounded below by f that approximates f with arbitrary accuracy.

Consequently we define the expectation of an everywhere-positive function as the expectation of this approximating simple function. Because we were careful to consider only simple functions bounded by f we can also define the expectation of f as the largest expectation of all bounded simple functions, E π [ f ] = max

For functions that aren’t everywhere-positive we can decompose X into a collection of neighborhoods where f is entirely positive, A^{+}_{n} , and entirely negative, A^{-}_{m} . In those neighborhoods where f is entirely positive we apply the above procedure to define \mathbb{E}_{\pi} [ f \cdot \mathbb{I}_{A^{+}_{n}}], while in the neighborhoods where f is entirely negative we apply the above procedure on the negation of f to define \mathbb{E}_{\pi} [ -f \cdot \mathbb{I}_{A^{+}_{n}}]. . Those regions where g vanishes yield zero expectation values and can be ignored. We then define the expectation value of an arbitrary function g as the sum of these contributions, \mathbb{E}_{\pi} [ f ] = \sum_{n = 0}^{\infty} \mathbb{E}_{\pi} [ f \cdot \mathbb{I}_{A^{+}_{n}}] - \sum_{m = 0}^{\infty} \mathbb{E}_{\pi} [ -f \cdot \mathbb{I}_{A^{-}_{m}}].

Formally this procedure is known as and is a critical tool in the more general of which probability theory is a special case.

2.4 Measurable Transformations

Once we have defined a probability distribution on a space, X , and a well-behaved collection of subsets, \mathcal{X} , we can then consider how the probability distribution transforms when X transforms. In particular, let f: X \rightarrow Y be a transformation from X to another space Y . Can this transformation also transform our probability distribution on X onto a probability distribution on Y , and if so under what conditions?

The answer is straightforward once we have selected a \sigma -algebra for Y as well, which we will denote \mathcal{Y} . In order for f to induce a probability distribution on Y we need the two \sigma -algebras to be compatible in some sense. In particular we need every subset B \in \mathcal{Y} to correspond to a unique subset f^{-1}(B) \in \mathcal{X} . If this holds for all subsets in \mathcal{Y} then we say that the transformation f is and we can define a distribution, \pi_{*} by \mathbb{P}_{\pi_{*}} [ B ] = \mathbb{P}_{\pi} [ f^{-1} (B) ]. In other words, if f is measurable then a self-consistent allocation of probability over X induces a self-consistent allocation of probability over Y .

One especially important class of measurable functions are those for which f(A) \in \mathcal{Y} for any A \in \mathcal{X} in addition to f^{-1}(B) \in \mathcal{X} for any B \in \mathcal{Y} . In this case f transforms not only a probability distribution on X into a probability distribution on Y but also a probability distribution on Y into a probability distribution on X . In this case we actually have one unique probability distribution that is just being defined over two different manifestations of the same abstract system. The two manifestations, for example, might correspond to different choices of coordinate system, or different choices of units, or different choices of language capable of the same descriptions. These transformations then serve as translations from one equivalent manifestation to another.

Measurable transformations can also be used to project a probability distribution over a space onto a probability distribution over a lower-dimensional subspace. Let \varpi: X \rightarrow Y be a that maps points in a space X to points in the subspace Y \subset X . It turns out that in this case a \sigma -algebra on X naturally defines a \sigma -algebra on Y and the projection operator is measurable with respect to this choice. Consequently any on X will transform into a unique on Y . More commonly we say that we the complementary subspace, Y^{C} .

Marginalization is a bit more straightforward when we are dealing with a product space, X \times Y , which is naturally equipped with the component projection operators \varpi_{X} : X \times Y \rightarrow X and \varpi_{Y}: X \times Y \rightarrow Y . In this case by pushing a distribution over (X \times Y, \mathcal{X} \times \mathcal{Y}) forwards along \varpi_{X} we marginalize out Y to give a probability distribution over (X, \mathcal{X}) . At the same time by pushing that same distribution forwards along \varpi_{Y} we can marginalize out X to give a probability distribution over (Y, \mathcal{Y}) .

Consider, for example, the three-dimensional space, \mathbb{R}^{3} , where the coordinate functions serve as projection operators onto the three axes, X , Y , and Z . Marginalizing out X transforms a probability distribution over X \times Y \times Z to give a probability distribution over the two-dimensional space, Y \times Z = \mathbb{R}^{2} . Marginalizing out Y then gives a probability distribution over the one-dimensional space, Z = \mathbb{R} .

2.5 Conditional Probability Distributions

As we saw in Section 2.4 , projection operators allow us to transform a probability distribution over a space to a probability distribution on some lower-dimensional subspace. Is it possible, however, to go the other way? Can we take a given marginal probability distribution on a subspace and construct a joint probability distribution on the total space that projects back to the marginal? We can if we can define an appropriate probability distribution on the complement of the given subspace.

Consider a N -dimensional , X , with the projection, \varpi : X \rightarrow Y , onto a K < N -dimensional , Y . By pushing a probability distribution on X along the projection operator we compress all of the information about how probability is distributed along the \varpi^{-1} (y) for each y \in Y . In order to reconstruct the original probability distribution from a marginal probability distribution we need to specify this lost information.

Every fiber takes the form of a N - K -dimensional space, F , and, like subspaces, these fiber spaces inherent a natural \sigma -algebra, \mathcal{F} , from the \sigma -algebra over the total space, \mathcal{X} . A defines a probability distribution over each fiber that varies with the base point, y , \begin{alignat*}{6} \mathbb{P}_{F \mid Y} :\; \mathcal{F} \times Y \rightarrow \; [0, 1] \\ (A, y) \mapsto \mathbb{P}_{F \mid Y} [A, y]. \end{alignat*} Evaluated at any y \in Y the conditional probability distribution defines a probability distribution over the corresponding fiber space, (F, \mathcal{F}) . On the other hand, when evaluated at a given subset A \in \mathcal{F} the conditional probability distribution becomes a measurable function from Y into [0, 1] that quantifies how the probability of that set varies as we move from one fiber to the next.

Given a marginal distribution, \pi_{Y} , we can then define a probability distribution over the total space by taking an expectation value, \mathbb{P}_{X} [ A ] = \mathbb{E}_{Y} [ \mathbb{P}_{F \mid Y} [A \cap \varpi^{-1} (y), y] ].

The induced joint distribution on the total space is consistent in the sense that if we transform it back along the projection operator we recover the marginal distribution with which we started.

This construction becomes significantly easier when we consider a product space, X \times Y and the projection \varpi: X \times Y \rightarrow Y . In this case the fiber space is just X .

The conditional probability distribution becomes \begin{alignat*}{6} \mathbb{P}_{X \mid Y} :\; \mathcal{X} \times Y \rightarrow \; [0, 1] \\ (A, y) \mapsto \mathbb{P}_{X \mid Y}[A, y]. \end{alignat*} with joint distribution \mathbb{P}_{X \times Y} [ A ] = \mathbb{E}_{Y} [ \mathbb{P}_{X \mid Y} [A \cap X, y] ].

Conditional probability distributions are especially useful when we want to construct a complex probability distribution over a high-dimensional space. We can reduce the specification of the ungainly joint probability distribution with a sequence of lower-dimensional conditional probability distributions and marginal probability distributions about which we can more easily reason. In the context of modeling an observational process, this method of construction a complicated distribution from intermediate conditional probability distributions is known as modeling. In particular, each intermediate conditional probability distribution models some fragment of the full observational process.

As we saw in the previous section, formal probability theory is simply the study of probability distributions that allocate a finite, conserved quantity across a space, the expectation values that such an allocation induces, and how the allocation behaves under transformations of the underlying space. While there is myriad complexity in the details of that study, the basics concepts are relatively straightforward.

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5 Minute Bookkeeping

The best tutorials, tips tricks to help you minimize bookkeeping and maximize profits

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One faithful day, Jeff opened his QuickBooks Online company file to view his Profit Loss report. He hadn’t looked through his QBO company in a while, but with tax time approaching, he wanted to make sure that nothing unusual stood out. But as Jeff looked through the Profit Loss report, he saw two strange accounts: Uncategorized Income and Uncategorized Expense . He couldn’t remember how these accounts even got there. He thought he was doing everything right, but now he wasn’t so sure. Jeff shuddered with disbelief – his tax accountant was not going to like this. Jeff knew that he had to take action, but he didn’t know where to begin. If you’re like Jeff, and you have uncategorized income and uncategorized expenses on your books, then stick around as I show you how to fix uncategorized income and expenses in QuickBooks Online.

Uncategorized Income Uncategorized Expense

One of the most common problems that I see on the Profit Loss report in QuickBooks Online are amounts in Uncategorized Income and Uncategorized Expense. You may not know that you have this problem. In fact, your tax accountant may bring it to your attention during tax time. If left unaddressed, your reports won’t be accurate, your accountant will be unhappy, and you’ll have to correct a lot of transactions!

Uncategorized Income and Uncategorized Expense.

Here is an example of what these accounts look like in the Profit Loss report:

Here is a video you can follow along:

You can easily find these uncategorized transactions by running a “quick report”. Simply go to the Left Navigation Bar ,

Left Navigation Bar Accounting Chart of Accounts “Uncategorized Income” “Uncategorized Expense” Run Report

A “quick report” window will open showing you all transactions in the account.

It all starts with downloaded banking transactions. When banking transactions are downloaded into QuickBooks Online, QBO can’t always figure out how to categorize the transaction. In that case, QBO assigns the U ncategorized Income account to amounts received and the Uncategorized Expense account to amounts paid.

To avoid this, you need to tell QuickBooks Online how to categorize the transactions. So, if you ever see Uncategorized Income in the “Category orMatch” column, you should select the appropriate category (or account from your chart of accounts).

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