rochfor1 said:There's a change of variables formula for two r.v.s for situations like this. It amounts to changing variables in the integral of the joint pdf.
A probability distribution function, also known as a probability density function, is a mathematical function that describes the probability of a random variable taking on a certain value or falling within a specific range of values. It is used to model the distribution of a continuous random variable.
The PDF of a continuous random variable is calculated by taking the derivative of the cumulative distribution function (CDF) of that variable. In other words, the PDF is the gradient of the CDF.
The use of arctan in a PDF allows for the creation of a skewed distribution, as opposed to a symmetrical distribution. This can be useful in modeling real-world data that may not follow a normal distribution.
The arctan function is typically used as a transformation within the PDF formula. For example, the arctan function can be used to transform a normally distributed random variable into a skewed distribution by applying it to the variable's CDF.
A PDF with arctan can be used in various fields, such as finance, biology, and physics. For example, in finance, it can be used to model stock prices, which often exhibit a skewed distribution. In biology, it can be used to model the distribution of body sizes in a population. In physics, it can be used to model the distribution of particle velocities in a gas.