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Thank you for the tasks

For the first part, I think there is no problem, because you just write the Poissonian formula.

The task was to find the mean number, the probability to have a single count if the

mean number of counts is 3.

So we just write p of m is 1 over m factorial and e to the minus mean number of counts,

mean number of counts to the power m, and you should substitute here m1 and here 3.

The difficulty might have appeared when you considered thermal light.

So for thermal light it will be a little bit more complicated.

You have to average with the intensity distribution.

So you have to average over dI with p of I, and here you have to substitute the Poissonian

formula, but with the mean number of photons being proportional to intensity.

And I want it to be very clear, proportional but not equal, because I'm sorry, but it's

very important that intensity is a dimensional variable.

It's measured in watts per square centimeter, for instance, or per square meter, and so

it cannot be equal to the number of photons.

So you have to write here mean number m as a function of I, yeah, to the power m, and

here e to the minus m of I, and here m of I, this mean number of photons, is just proportional

to the mean intensity.

So it has some, and this we derived at the last lecture, that the mean number of photons

is always proportional to the mean intensity.

But this coefficient alpha, it's a dimensional coefficient.

So if the intensity is in watts per square meter, for instance, then alpha in square

meter per watt.

And then if you plug it, you will get p of m is, well, 1 over m factorial.

You can drag out of the formula.

Then there is d of dI.

p of I, we know it's 1 over mean intensity e to the minus intensity over mean intensity.

And then here you have alpha mean intensity to the power m, and e to the minus alpha mean

intensity.

And then, of course, from this condition, you can derive that alpha is 3 over mean intensity

for our case, because we know that the mean number of counts is 3.

Yes?

Yes?

I have a question about this, because when the mean intensity is just a value, so it

does not vary, so we can pull it actually out of the integrand.

Of course.

Yeah, yeah, we can.

So this would give us the same formula as above for the constant intensity.

Wait a minute, wait a minute.

Sorry, sorry, sorry, sorry, sorry, sorry.

No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no,

no, no, no, no, no.

I just did a mistake.

Yeah, it's this, this way.

Yeah, you're right.

Yeah, you just caught me.

Yeah, yeah.

It's not a mean intensity.

It's just this.