Probability distribution: exponential of a quartic

In summary, a probability distribution is a mathematical function that maps the possible outcomes of a random variable to their associated probabilities. An exponential distribution is a type of probability distribution that describes the time between events in a Poisson process. In the context of probability distributions, "quartic" refers to the fourth power or degree. The exponential distribution of a quartic function is calculated by taking the fourth power of the random variable and multiplying it by the exponential constant (e) raised to the power of the negative of the fourth power of the random variable. This function is commonly used in various fields such as physics, biology, economics, finance, and insurance to model and calculate probabilities.
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vvaibhav08
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I've been trying to work out the normalization of this probability distribution but have found very few resources that have dealt with this. (one being: https://www.tandfonline.com/doi/abs/10.1080/00401706.1978.10489702). Can anyone help me in solving this? Additionally, I also wanted to know the astronomical relevance of this distribution (Where in astrophysics/cosmology does this distribution come up?)

Any help is much appreciated.
$$\int_{-\infty}^{+\infty} exp (-[ax + bx^2]^2) dx$$
$$a\&b\in R$$
 
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  • #2
Well, doing some change of variables (a shift to complete square and then define the exponent as ##x##) I've reduced the integral to $$\int_{-\frac{a^2}{4b}}^{\infty}\frac{e^{-x^2}}{\sqrt{bx+\frac{a^2}{4}}}\text{d}x$$ Which seems to be solvable using the Bessel functions, unfortunately, the solution that WolfralAlpha give is a complex number, so I don't know how to solve it, maybe from here you can work something more.
 
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FAQ: Probability distribution: exponential of a quartic

1. What is a probability distribution?

A probability distribution is a mathematical function that describes the likelihood of different outcomes occurring in an experiment or event. It shows the possible values of a random variable and their associated probabilities.

2. What is an exponential distribution?

An exponential distribution is a type of probability distribution that describes the time between events in a Poisson process, where events occur continuously and independently at a constant average rate. It is often used to model waiting times or survival times.

3. What is a quartic function?

A quartic function is a polynomial function of degree four, meaning it has the form f(x) = ax^4 + bx^3 + cx^2 + dx + e. It is also known as a fourth-degree function and can have up to four real roots.

4. How is the exponential of a quartic used in probability distribution?

The exponential of a quartic is used in probability distribution to model the probability of a quartic function taking on different values. This can be useful in situations where the data follows a quartic pattern or when the data has a high degree of variability.

5. What is the relationship between the exponential and quartic distributions?

The exponential distribution is the continuous analog of the geometric distribution, while the quartic distribution is a type of polynomial distribution. The exponential of a quartic combines these two distributions and can be used to model data that has both exponential and quartic characteristics.

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