Discrete Distributions (Alatissa expert in probability theory)

I will now show the table in full, as it turned out in Excel calculations



Here we paid attention to the exponential notations and the fact that the calculations are up to nine decimal places.
That is, rounding occurs, and small errors result.
It's actually a bit early to get a value of one at step seven.

If anyone is interested, I will show you a table where I have already converted exponential notations to decimal notations.
It will look like this

Next, I will do the calculations when the values of the distribution function are 14 decimal places
and it will be more clear what these meanings are
i.e. Excel's automatic calculations somehow didn't really suit me)



Next, for what purposes is the fr calculated?
What questions can be answered when there are calculations of the distribution function and probability density

in fact, excel calculations of norms are just very rounded.


...



Anyone who looked closely at the column with the calculations of the integral function understood what they are:
f(0)= p(0)
f(1) = p(0) + p(1)
f(2) = p(0) + p(1) + p(2)
f(3) = p(0) + p(1) + p(2) + p(3)
f(4) = p(0) + p(1) + p(2) + p(3) + p(4)
f(5) = p(0) + p(1) + p(2) + p(3) + p(4) + p(5)
etc.

That is, the probabilities are summed up, and each time it gets closer and closer to one.

p(0) + p(1) + p(2) + p(3) + p(4) + p(5) + p(6) + p(7) + p( + p(9) + p(10) = 1

Therefore, in automatic calculations, many units will often be encountered in the values of the integral fr, these are simply values very close to one and, according to the rounding rule, are taken as 1.


Also in automatic calculations, values in exponential notation will often be encountered, and E - in
For example, the number 5.97e -15 means 5.97 multiplied by 10 to the power of -15, where 5.97 is the number a and -15 is the number b.
That is, values very close to zero.

Next, when there are already calculations of both pv and fr

Random variable in binomial distribution - number of times.

There are a lot of problems on inequalities in the theory of inequalities,
i.e. find the probability that a random variable will fall within a certain interval.
For example, in this example with a limit of 10 spins, the wording could be:
Find the probability that the number will be repeated:
less than 4 times,
at least 3 times,
no more than 3 times,
more than 2 times,
at least once,
more than one and no more than 3 times
and others

That is, we are talking about intervals here.

Less than 4 times is when x2) = p(3) + p(4) + p(5) + p(6) + p(7) + p( + p(9) + p(10)

At least once is when x>0
probability (x>0) = 1- p(0)

More than one and no more than 3 times
This is a double inequality 1

Next, the mathematical expectation for the binomial distribution.