Separating the product of two probability distributions

In summary: This will give you the marginal distribution of r, which is what you are looking for. In summary, to separate the product of two probability distributions with one of them known, you can set it up as P(r<R) = ∫₀¹ P(w<Rcosθ|cosθ) d(cosθ). After integration, this will give you the marginal distribution of r, which is what you are looking for.
  • #1
pboggler
2
0
In general, how does one separate the product of two probability distributions with one of them known? Basically, I have the distribution of rcosθ, I know that P(cosθ) = 2/(πsinθ), and I want to find P(r). Wolfram Alpha makes me think that a delta function is involved based on what they say about uniform product distributions and normal product distributions, but I wouldn't know how to solve it in this case.

Here are the URLs to Wolfram Alpha's things. I can't include links until I make 10 posts apparently.
mathworld.wolfram.com/UniformProductDistribution.html
mathworld.wolfram.com/NormalProductDistribution.html

Thanks for any help or suggestions!
 
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  • #2
Assuming r and θ are independent, you can set it up as follows (admittedly sketchy):

Let w = rcosθ, then P(r < R|cosθ) = P(w<Rcosθ|cosθ). Integrate over the distribution of θ.
 
  • #3
What should I be integrating? And what will it tell me? Assuming the distributions are over [0,1], are you saying something like [itex]P(w) = \int_0^1 P(w<Rcosθ|cosθ) d(cosθ)[/itex]?
 
  • #4
pboggler said:
What should I be integrating? And what will it tell me? Assuming the distributions are over [0,1], are you saying something like [itex]P(w) = \int_0^1 P(w<Rcosθ|cosθ) d(cosθ)[/itex]?

After integration you will end up with P(r<R).
 
  • #5


I would approach this problem by first understanding the concept of probability distributions and how they can be manipulated. In general, the product of two probability distributions is not a well-defined concept, but it can be approximated in certain cases.

In this specific case, we have the distribution of rcosθ and we know that P(cosθ) = 2/(πsinθ). To find P(r), we need to manipulate this expression to get it in terms of r. One approach could be to use the definition of a probability distribution, which states that the integral of the probability density function (PDF) over the entire domain must equal 1. So, we can set up an integral for P(r) as follows:

∫P(r)dr = ∫P(rcosθ)dr

Now, we can use a change of variables to express the integral in terms of r. Let u = rcosθ, then du = cosθdr. Substituting this into the integral, we get:

∫P(r)dr = ∫P(u)du/cosθ

Since we know P(u) = 2/(πsinθ), we can substitute this in and solve for P(r):

∫P(r)dr = ∫2/(πsinθ)du/cosθ

= 2/π∫du/(sinθcosθ)

= 2/π∫du/sin(2θ)

= 1/π∫du/sinθ

= 1/π∫dr/r

= 1/π ln(r) + C

Therefore, we have found the expression for P(r) in terms of r. This is not a delta function as suggested by Wolfram Alpha, but a logarithmic function. This approach is just one possible way to separate the product of two probability distributions. Depending on the specific case, there may be other techniques that could be used.

In general, separating the product of two probability distributions with one of them known can be a challenging task and may require advanced mathematical techniques. It is important to carefully consider the properties of the distributions and use appropriate methods to manipulate the expressions in order to find a solution. I would recommend consulting with a statistician or a mathematician for further assistance in solving this type of problem.
 

1. Can you explain the concept of separating the product of two probability distributions?

Separating the product of two probability distributions refers to the process of breaking down a joint probability distribution into its individual components. This allows for a better understanding of the relationship between the two distributions and how they contribute to the overall probability.

2. Why is separating the product of two probability distributions important?

Separating the product of two probability distributions is important because it helps to identify the independent variables that contribute to the overall probability. This can aid in making more accurate predictions and understanding the underlying factors that influence the outcome.

3. What are some common methods used to separate the product of two probability distributions?

Some common methods used to separate the product of two probability distributions include the Law of Total Probability, Bayes' Theorem, and the Chain Rule. These methods involve breaking down the joint distribution into conditional probabilities and using mathematical operations to separate the variables.

4. How does separating the product of two probability distributions relate to statistical modeling?

Separating the product of two probability distributions is a fundamental concept in statistical modeling, as it allows for a better understanding of the relationship between variables and their impact on the overall probability. This is essential for developing accurate and reliable statistical models.

5. Can separating the product of two probability distributions be applied to real-world scenarios?

Yes, separating the product of two probability distributions can be applied to real-world scenarios. For example, it can be used in risk analysis to identify the factors that contribute to a particular outcome, or in finance to understand the relationship between different variables and their impact on investment decisions.

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