Derive Analytic Formula for CDF - Homework Equations & Solution

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In summary, the conversation is about deriving an analytic formula for a CDF and understanding the definition of an analytic formula. The suggested formula for the CDF is F(x) = \int_{-\infty}^x f(t)\, dt \hbox{ so } F'(x) = f(x), and it is confirmed that this is what the person asking the question meant. The conversation then shifts to a different topic, asking for help with a problem involving proving that if f(z) is analytic, f*(z*) is also analytic.
  • #1
patm95
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Homework Statement



Derive an analytic forumula for a CDF

Homework Equations





The Attempt at a Solution



All I really want to know is what an analytic formula is? I know that the CDF is basically an integral of the PDF. I have looked all over the internet and can not find anything of the definition of an analytic formula.

Thanks!
 
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  • #2
Unless they have a specific CDF F(x) in mind, perhaps they just mean

[tex]F(x) = \int_{-\infty}^x f(t)\, dt \hbox{ so } F'(x) = f(x)[/tex]

Edit Corrected typo.
 
Last edited:
  • #3
Thanks. That's what I kinda figured, but I wanted to make sure that is what they ment.

Thanks again!
 
  • #4
Hi can someone help me please? I am trying to do the following problem:
Prove that if f(z) is analytic, f*(z*) is also analytic. The stars represent conjugates. Thanks!
 

1. What is the purpose of deriving an analytic formula for a cumulative distribution function (CDF)?

The CDF is a mathematical function that describes the probability of a random variable being less than or equal to a certain value. Deriving an analytic formula for the CDF allows us to calculate these probabilities without having to rely on tables or numerical methods, making it easier to analyze and interpret data.

2. What are the necessary equations and formulas needed to derive an analytic formula for a CDF?

To derive an analytic formula for a CDF, you will need to have a basic understanding of probability theory and calculus. The main equations and formulas you will use include the definition of the CDF, the probability density function (PDF), and the fundamental theorem of calculus.

3. How do you approach deriving an analytic formula for a CDF?

The first step in deriving an analytic formula for a CDF is to define the random variable and determine its probability density function. Then, you can use the fundamental theorem of calculus to integrate the PDF and obtain the CDF. Finally, you may need to apply any necessary transformations or substitutions to simplify the formula.

4. Can an analytic formula for a CDF be derived for any probability distribution?

Not all probability distributions have a closed-form analytic formula for their CDF. For some distributions, it may be impossible to derive a closed-form solution, and numerical methods may be required to calculate the CDF. However, many common distributions, such as the normal distribution and the exponential distribution, have well-known analytic formulas for their CDFs.

5. What are the benefits of using an analytic formula for a CDF over other methods for calculating probabilities?

Using an analytic formula for a CDF can provide a more accurate and precise calculation of probabilities compared to using tables or numerical methods. It also allows for easier interpretation and analysis of data, as the formula can be used to calculate probabilities for different values of the random variable. Additionally, once the formula is derived, it can be used repeatedly without the need for additional calculations, saving time and effort.

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