## QM: G1.3 Probability and Expectation

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### QM: G1.3 Probability and Expectation

In the discrete case:

let j be the number of samples, and N(j) the population of objects found in each sample, there is the trivial case where the population of objects per sample is just 1 so N(j) = for all j that have objects and N(j) = j for all j without objects. The formalism is a little difficult like all formalisms will usually be.

The total number of objects for all samples:

For the non trivial case

But it gets a little unwieldy in the continuous case. So I'll work through it a bit.

Suppose we have a clock that goes form 0 to 5 seconds. What is the expected value of all possible measurement of time in that time frame? Well, a discrete approximation is just taking the the time readings from start to finish at 1 second intervals:

j = 0, 1, 2, 3, 4, 5 and N(j) = 1 for all j

Using the above formula (0+1+2+3+4+5) * (1/6) = 15/6 = 2.5

Before going into the formalism let's just try to do this in the continuous case where we run the clock from 0 to 5 seconds. What is the average time (which ONLY in the special case the average is also the expectation value). I will use Psi notation to drive the point home a bit. I begin with the Born constraint (which is really true of all probability conceptions):

Now, by inspection and assumption, the probability density P(t) is independent of t, so we can, thankfully, solve this integral:

Which implies

for all t.

So now we can state the definition of expectation in the continuous case and then solve and example:

So what is the expected value of t as we run the clock from 0 to 5 seconds?

So bear in mind, even for trivial problems the formal rigor can feel like rigor mortis!
Last edited by stcordova on Mon Feb 12, 2018 1:39 pm, edited 2 times in total.
stcordova

Posts: 447
Joined: Wed Mar 05, 2014 1:41 am

### Re: QM: G1.3 Probability and Expectation

useful little theorem on variances:

stcordova

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Joined: Wed Mar 05, 2014 1:41 am

### Re: QM: G1.3 Probability and Expectation

The continuous case theorem: slide_49.jpg (52.68 KiB) Viewed 1710 times
stcordova

Posts: 447
Joined: Wed Mar 05, 2014 1:41 am

### Re: QM: G1.3 Probability and Expectation

Incidentally:

stcordova

Posts: 447
Joined: Wed Mar 05, 2014 1:41 am 