## Binomial distribution – Critical values ​​/ Critical region: Statistics and mechanics Edexcel Year 1 AS: p.104 Ex.7B Q5.

I do not agree with the answer they give.

P (X = 0) = 0.0188 …> 0.01 = 1%. Let's go back to the definition of "critical region" and compare to the example of page 101/102 (which I will not publish here for the sake of brevity):

Now, it would be "surprising / unlikely" to get 0 success considering X ~ B (20.0.18). However, 1.88% is not "less than 1% (not) likely", and so if we performed these 20 tests and achieved 0 successes, it is certain that we would not reject the null hypothesis at the 1% significance level.

Am I right or is there a flaw in my reasoning?

## probability – How to find the \$ E (X) \$ value of a direct integration hypergeometric distribution?

I am curious to know how to find the expected value of the hypergeometic distribution by direct integration?

I know it $$Gamma left (x right) = int limits_0 ^ infty {s ^ {x – 1} e ^ {- s} ds}$$is it still linked to the solution of the expected value of $$E (x)$$

Another sub-question arises, what we integral to $$x!$$and what are these defined limits, I know that there may be no closed-form solution to this integral with the current list of special functions.

I've googled and checked many references on the integrals, including the CRC Handbook of Integrals, to no avail :(.

Here is my attempt at the question I asked!

-C & # 39; is my attempt

I know that there is a way to find it by direct integration and without any other means. Will an approach to mass representation work?

## Irwin-Hall distribution relationship between two sets of events

Let $$X$$, $$Y$$, $$Z$$, $$A$$ to be a set of random variables derived from the Irwin-Hall distribution where $$X$$ is the sum of $$c$$ iid r.v.s, $$Y$$ is the sum of $$c$$ iid r.v., $$Z$$ is the sum of $$n – c$$ iid r.v.s, and $$A$$ is the sum of $$n – c$$ iid r.v.s.

I want to compare $$Pr[(X leq x) cap (Z leq x – X) cap (A leq x – X)]$$ with $$Pr[(X leq x) cap (Y leq x) cap (Z leq x – X) cap (A leq x – Y)]$$. Intuitively, it seems that $$Pr[(X leq x) cap (Z leq x – X) cap (A leq x – X)] geq Pr[(X leq x) cap (Y leq x) cap (Z leq x – X) cap (A leq x – Y)]$$ but I could not find proper proof for that.

You can assume that $$x in [0, n]$$.

## probability or statistics – How to define a discrete distribution with non-integer sample space elements (results)?

I know how to define a discrete distribution with integer states:

``````        state: 1 2 3
P[X==state]   0.3 0.4 0.3
``````

I want to define a discrete distribution with non integer states:

``````        State: 0.01 0.02 0.03
P[X==state]   0.3 0.4 0.3
``````

It is possible to use `By pieces[]` and `Probability distribution[]` define a distribution with integer states.

``````pmf[x_] : = By pieces[{
{0.3, x == 1}
, {0.4, x == 2}
, {0.3, x == 3}
}];
distribution = ProbabilityDistribution[pmf[x], {x, 1, 3, 1}

;
``````

### But `Probability distribution[]` seems unable to work with non-integer states (it does not even work with integer states with $$dx = 2$$)

Is it a bug, a feature or a convention?

## Question.

How to define the distribution of non-integer states (above)?

``````pmf[x_] : = By pieces[{
{0.3, x == 0.01}
, {0.4, x == 0.02}
, {0.3, x == 0.03}
}];
distribution = ProbabilityDistribution[pmf[x], {x, 0.01, 0.03, 0.01}];

Probability[X > .02, X [Distributed] Distribution]
``````

## st.statistics – The average E (X) of the negative binomial distribution

What I know about the mean of the negative binomial distribution is E (x) = r (1-p) / p. but there are some questions use E (x) = r / p as mean. Very confusing and I do not understand at all.

For example:

Regularly roll a die just until result 3 occurs for the 4th time.
Let X be the number of times needed to achieve this goal. Find
E (X) and Var (X)?
My answer: negative pair with r = 4, p = 1/6. E (x) = r (1-p) / p = 20
However, the correct answer is: E (x) = r / q = 24

and for this question:
The probability that a basketball player makes a free kick is 60%.
The player was asked not to leave the practice unless he made 10 shots. Let Y be
the number of free throws missed before the 10th throws. Find the average
and the variance of Y.
My answer is right. Negative binomial with r = 10, p = 0.6. E (y) = r (1-p) / p = 6.6.67

I do not understand why there are 2 formulas and how to tell the difference, which one should I use?

## algorithms – Approximation of a discrete distribution

I have a discreet reference distribution. For example, say:

• P (X = 1) = 0.2
• P (X = 2) = 0.7
• P (X = 3) = 0.1

Now I am given `not` numbers, and I want to group (summarize) these numbers in 3 classes and approach as much as possible the above distribution in the sense of minimizing the sum of the squared error. So let's say that I have these numbers: 10, 25, 25 50 (total sum = 100). So, I want to group them in 3 bins, and ideally the sum of each bin would be 20, 70, 10 and that would be perfect for the distribution. Unfortunately, this is not possible and the best here would be 25, 75 (50 + 25), 10. The error here is (25-20) ² + (75-70) ² + (10- 10) ² = 50

What does the algorithm solve the general problem?

## Functional Analysis – A String Rule in the Distribution Space \$ mathcal {D} & # 39; \$

Many calculation results are valid for distributions. In any case, I did not find anything on a string rule for derivatives of distributions (obviously, when they are of the function type, hence elements of $$L ^ 1_ {loc} subset mathcal {D}$$).

A non-technical but interesting result?

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## Distribution support

How the support of the Dirac distribution is {0}.
I started reading the distributions a few days ago.
Someone can help me?

## hypothesis test – likelihood ratio, distribution of the test statistic

I want to perform a statistical test based on two simple assumptions. I have the analytical form of my pdf based on 3 parameters that I know are set to alpha = 0.65, beta = 0.06, gamma = -0.18 for a hypothesis and alpha = 1/3, beta = 0, gamma = 0 for the other hypothesis. For the second hypothesis, the pdf is reduced to a uniform pdf.

I have a data set of 50,000 points sampled from the first distribution (the nonuniform), so the test is trivial but I still have to do it.

I calculate the value of each pdf at each point of the dataset, take their logarithm, then subtract them and add the result to get the log (lambda). Now, I find a very weak result, which means exp (-40772), which clearly supports the non-uniform distribution of data, but how can I calculate the region of significance or the power of the test? I can not use Wilk's theorem because my pdf is completely defined. The only thing I can do is:

P (lambda <c | H0) = alpha

But I do not know how to calculate c. Anyone have any suggestions?

For clarity, I will post both PDF files:

variables $$theta, phi$$ defined in $$theta in [0, pi] phi in [0, 2pi]$$.

$$(θ, φ) = 34π[0.5(1−𝛼)+(0.5)(3𝛼−1)𝑐𝑜𝑠(𝜃)2−𝛽𝑠𝑖𝑛(𝜃)2𝑐𝑜𝑠(2𝜙)−2√𝛾𝑠𝑖𝑛(2𝜃)𝑐𝑜𝑠(𝜙)]$$

$$alpha = 0.65, beta = 0.06, gamma = -0.18$$

Uniform pdf:
$$U ( theta, phi) = begin {case} frac {1} {2 pi ^ {2}} & theta in [0, pi] phi in [0, 2pi]\ 0 & otherwise end {cases}$$