Expectation values and mean square deviations

In summary, the conversation revolves around computing the expectation value and mean square deviation for a die with six possible states. The discussion covers the definition of expectation, the difference between theoretical and statistical values, and the use of probability and statistics in quantum theory. The calculation for the expectation value is shown to be 3.5 by using the distributive law, while the value for <s^2> is 15.166... The concept of an X value such that EX^2 < (EX)^2 is also brought up and further explanation is requested.
  • #1
Ed Quanta
297
0
I am having trouble applying this concept to simple things. Like a die where we let s be the number of spots shown by a die thrown at random.



How could I compute the expectation value of s? And how would I compute the mean square deviation. Would the expectation value just be <s>=1/N summation from i=1 to N of si? Here the only possible states are s1,s2,s3,s4,s5, and s6. Help anyone? And how would I calculate the mean square deviation from the equation delta s=<<s-<s>>^2>=<c^2>-<c>^2?
 
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  • #2
Would the expectation value just be <s>=1/N summation from i=1 to N of si?

Yes, but you seem to be unsure... do you recall the definition of expectation?

<c^2>-<c>^2?

(assuming you mean <s^2> - <s>^2)
You've already got <s>, you just need <s^2>, which you can also compute from the definition.
 
  • #3
Yeah, I am unsure because I always thought that the expectation value of let's say <x> was the integral of the wave functions squared multiplied by x. But the definition I gave was not in integral form. Does the integral form only have to do with continuous states, where as for the die there are only 6 different possible states? sorry if I am being confusing or just making more of this than there is. Oh and one more question, the only specification of N is that it is much larger than 1. Is this simply because probablity of a system of randomness will be more and more accurate as N approaches infinity?
 
  • #4
There is a distinction between theoretical values, which can be computed directly, and statistical values. To compute the theoretical value, assume each face of a die has probability 1/6, then the expectation values are <s>=3.5, and <s2>=15.166... The statisitcal value is obtained by running lots of trials and averaging the results by the number of trials.

As an afterthought, these concepts have nothing to do with quantum theory, although quantum theory uses them. These are notions from probability and statistics.
 
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  • #5
How did you get those values? Did you just add 1+2+3+4+5+6 and divide by 6 to calculate <s>? And if so, why would this be accurate for calculating an expectation value where 3.5 be the expectation value when there is an equal probability for each of the six states?
 
  • #6
Distributive law.

1 * 1/6 + 2 * 1/6 + 3 * 1/6 + 4 * 1/6 + 5 * 1/6 + 6 * 1/6 = (1 + 2 + 3 + 4 + 5 + 6)/6
 
  • #7
Why is Expectation<s^2> = 15.1666?

Is it possible for there to be an X such that EX^2 < (EX)^2? Please explain.
 
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1. What is an expectation value in statistics?

An expectation value, also known as an average or mean, is the sum of all the possible outcomes of a random variable multiplied by their respective probabilities. It represents the most likely outcome of a random event.

2. How is the expectation value calculated?

The expectation value is calculated by multiplying each possible outcome by its probability and then summing all of these values together. This can be expressed mathematically as E(X) = ∑x * P(x), where x represents the possible outcomes and P(x) represents their respective probabilities.

3. What is the significance of the expectation value in statistics?

The expectation value is an important measure in statistics because it represents the central tendency of a data set. It provides a single value that summarizes the data and can be used to make predictions about future events.

4. What is the difference between expectation value and mean square deviation?

The expectation value and mean square deviation are two measures of central tendency in statistics. The expectation value represents the most likely outcome, while the mean square deviation measures the spread or variability of the data around the expectation value. In other words, the mean square deviation tells us how much the data points deviate from the expectation value.

5. How can expectation values and mean square deviations be used in practical applications?

Expectation values and mean square deviations are commonly used in various fields, such as finance, physics, and engineering, to make predictions and analyze data. For example, in finance, these measures can be used to calculate the expected return and risk of an investment. In physics, they can be used to predict the behavior of particles in quantum mechanics. In engineering, they can be used to analyze the reliability and performance of systems.

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