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This phrase came from the book Understanding Probability written by Henk Tijms. The original sentence is:

the probability that a normally distributed random variable will take on a value that lies z or more standard deviations above the expected value is equal to 1 − Φ(z) for z > 0, as is the probability of a value that lies z or more standard deviations below the expected value.

I don't understand what lies means in this context. I have checked The Oxford Advanced Learner's Dictionary, and Webster's online; none of their explanations refer lie as a transitive verb. How can it used with z or more standard deviations, which is obviously a noun phrase?

Could anyone explain to me:

  1. What does the sentence mean?
  2. The grammar underlying this sentence? Am I right that the lie used here is actually a transitive verb?
  • 1
    I have added the lower case phi to your quotation. If φ is not the correct phi, please copy and paste the appropriate version in: ϕ Φ . If you wish to find special characters not on your keyboard in the future, you can use the program character map (Windows and Ubuntu have this; I don't know about Macs, other Linux distros or mobile devices) or search for the name of the character, then copy and paste. – Esoteric Screen Name Aug 20 '14 at 11:15
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    It's the uppercase one, I reedit the question, thank you. – Rui Aug 20 '14 at 13:25
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In this case, z or more standard deviations is a measure of distance. You're right about its composition but not about its usage; it's a noun phrase but lies is still intransitive. The phrase quantifies above the expected value rather than functioning as the object of lies. The entire clause z or more standard deviations above the expected value acts as an adverb to modify lies; it tells us precisely where the value lies.

Here's a break down and subsequent simplification.

[A value that]1 [lies]2 [z or more standard deviations]3 [above the expected value]4.

  1. The subject; let's simplify by replacing this with the city.
  2. The verb.
  3. How far; let's simplify by replacing this with 50 kilometres.
  4. Where; let's simplify by replacing this with away.

Now, the sentence becomes:

The city lies 50 kilometres away.

The grammar is exactly the same. 50 kilometres away is an adverbial clause telling us where the city lies.

1

Of the Oxford Advanced Learner's Dictionary's definitions, the fourth one is most applicable:

4 [intransitive]
+ adverb/preposition
(of a town, natural feature, etc.) to be located in a particular place
The town lies on the coast.

When the values and distribution are plotted, such as on a histogram, a value can be said to lie somewhere on the plot. The use of lie implies some kind of visualization.

You can parse the sentence as

  • Subject: The probability
    • Subordinate clause: that
      • Subject: a normally distributed random variable
      • Verb: will take on
      • Object: a value
      • Subordinate clause: that
        • Verb: lies
        • Adverbial phrase: z or more standard deviations
        • Prepositional phrase: above the expected value
  • Verb: is
  • Adjective: equal
  • Prepositional phrase: to 1 − φ(z)
  • Prepositional phrase: for z > 0

The verb lie is intransitive: its complement is a prepositional phrase ("above the expected value").


From a technical viewpoint, the sentence is much more complex than it needs to be. I would probably rephrase it as

The probability that a normally distributed random variable will take on a value ≥ z, where z is the number of standard deviations above the expected value, is 1 − φ(z). By symmetry, the probability of a value z or more standard deviations below the expected value is also 1 − φ(z).

  • I realize I'm nitpicking semantics, but your amended version changes the meaning of the calculation with the statement random variable will take on a value ≥ z (only in the first sentence; the second matches the original's meaning). The random variable is a data point and not bounded by or compared to z, which is the number of standard deviations. Consider the simple case with expected value = -1000, z = 1, φ(1) = 0.5, standard deviation = 2. P(RV ≥ 1) [your statement] is effectively zero; P(RV ≥ -998) [the original] = 0.5. – Esoteric Screen Name Aug 20 '14 at 12:10
  • Thank you, It's really difficult to decide which answer to accept, but I had an vague understanding on the sentence already, and Esoteric's answer is more about the grammar, so I'll accept his answer. – Rui Aug 20 '14 at 13:43

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