Expectation values and mean square deviations

[Ed Quanta]

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?

 

[Hurkyl]

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.

 

[Ed Quanta]

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 probability of a system of randomness will be more and more accurate as N approaches infinity?

 

[mathman]

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.

 

[Ed Quanta]

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?

 

[Hurkyl]

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

 

[Nireus]

Why is Expectation<s^2> = 15.1666?

Is it possible for there to be an X such that EX^2 < (EX)^2? Please explain.