Coordination of Conjunctions and Punctuation

Coordination of Conjunctions and Punctuation Coordination of Conjunctions and Punctuation Coordination of Conjunctions and Punctuation By Mark Nichol When a conjunction is inserted into a sentence to separate two cumulative elements of the sentence, where commas, if any, are correctly positioned depends on the syntactical structure of the sentence regardless of whether a parenthetical phrase complicates the sentence. In each sentence with parenthesis below, the punctuation is not appropriate for the syntax. Discussion after each example explains the problem, and a revision provides a solution. 1. That debate could place everything on the table and, for that reason, significant tax reform in 2017 may prove challenging to achieve. This sentence consists of two independent clauses interrupted by the modifying phrase â€Å"for that reason† (which introduces the second clause). Without that phrase, the sentence would read, â€Å"That debate could place everything on the table, and significant tax reform in 2017 may prove challenging to achieve.† In the original sentence, â€Å"for that reason† is treated as a parenthetical phrase and is therefore bracketed by commas, but it is an introductory phrase, and so only the following comma is necessary: â€Å"That debate could place everything on the table, and for that reason, significant tax reform in 2017 may prove challenging to achieve.† 2. The business recently acted on the recommendation, and early on in its transformation process, has already generated valuable time and money-saving efficiencies. Here, the second part of the sentence shares the subject â€Å"the business,† so that section of the sentence is not an independent clause. The root sentence is â€Å"The business recently acted on the recommendation and has already generated valuable time and money-saving efficiencies.† Therefore, the punctuation should frame the parenthesis: â€Å"The business recently acted on the recommendation and, early on in its transformation process, has already generated valuable time and money-saving efficiencies.† 3. We observed several cases in which models were built solely based on a quantitative approach, and, as a result, generated poor model fit and model performance. This example has the same syntactical structure as the previous one but includes both a comma intended to separate independent clauses and a pair of commas to set off the parenthetical. However, the part of the sentence following the parenthetical is not an independent clause, so the first comma is an error: â€Å"We observed several cases in which models were built solely based on a quantitative approach and, as a result, generated poor model fit and model performance.† Want to improve your English in five minutes a day? Get a subscription and start receiving our writing tips and exercises daily! Keep learning! Browse the Punctuation category, check our popular posts, or choose a related post below:7 Examples of Passive Voice (And How To Fix Them)"Gratitude" or "Gratefulness"?Advance vs. Advanced

The Formula for Expected Value

The Formula for Expected Value One natural question to ask about a probability distribution is, What is its center? The expected value is one such measurement of the center of a probability distribution. Since it measures the mean, it should come as no surprise that this formula is derived from that of the mean. To establish a starting point, we must answer the question, What is the expected value? Suppose that we have a random variable associated with a probability experiment. Lets say that we repeat this experiment over and over again. Over the long run of several repetitions of the same probability experiment, if we averaged out all of our values of the random variable, we would obtain the expected value.   In what follows we will see how to use the formula for expected value. We will look at both the ​discrete and continuous  settings and see the similarities and differences in the formulas.​ The Formula for a Discrete Random Variable We start by analyzing the discrete case. Given a discrete random variable X, suppose that it has values x1, x2, x3, . . . xn, and respective probabilities of p1, p2, p3, . . . pn. This is saying that the probability mass function for this random variable gives f(xi)   pi.   The expected value of X is given by the formula: E(X) x1p1 x2p2 x3p3 . . . xnpn. Using the probability mass function and summation notation allows us to more compactly write this formula as follows, where the summation is taken over the index i: E(X)   ÃŽ £ xif(xi). This version of the formula is helpful to see because it also works when we have an infinite sample space. This formula can also easily be adjusted for the continuous case. An Example Flip a coin three times and let X be the number of heads. The random variable X  is discrete and finite.  The only possible values that we can have are 0, 1, 2 and 3. This has probability distribution of 1/8 for X 0, 3/8 for X 1, 3/8 for X 2, 1/8 for X 3. Use the expected value formula to obtain: (1/8)0 (3/8)1 (3/8)2 (1/8)3 12/8 1.5 In this example, we see that, in the long run, we will average a total of 1.5 heads from this experiment.  This makes sense with our intuition as one-half of 3 is 1.5. The Formula for a Continuous Random Variable We now turn to a continuous random variable, which we will denote by X.  We will let the probability density function of  X  be given by the function f(x).   The expected value of X is given by the formula: E(X)   Ã¢Ë† « x f(x) dx. Here we see that the expected value of our random variable is expressed as an integral.   Applications of Expected Value There are many applications for the expected value of a random variable. This formula makes an interesting appearance in the St. Petersburg Paradox.