# Pdf of Difference of Random Variables

• I
• marcusl
In summary, the conversation discusses finding the probability density function of the difference of two random variables, given that one of them is a sine function and the other is uniformly distributed. Various approaches were discussed, including using convolution and characteristic functions, but the conversation ultimately concludes that numerical integration may be the most efficient way to calculate the pdf.
marcusl
Gold Member
I want to find the probability density function (pdf) of the difference of two RV's,
$$p_{\Delta Y} = p_{(Y_1 - Y_2)},$$where $y = \sin \theta$, and where $\theta_1$ and $\theta_2$ are random variables with the same uniform distribution $p_{\theta}=\mathrm{rect}\left(\frac{\theta}{\pi}\right)$. This has support $-\pi/2\leq\theta\leq \pi/2$. (Please let me know if I am misusing terminology, as Math is not my native language )

I can derive the distribution of y $$p_Y=\frac{\mathrm{rect}\left(\frac{y}{2}\right)}{\pi \sqrt{1-y^2}}$$by starting from the uniform distribution. This is nonzero over $|y| \leq 1$. I get bogged down with the pdf of Δy, however. A direct approach involving the convolution$$p_{\Delta Y}(\Delta y)=p(y)*p(y)=\int_{-\infty}^{\infty}p_Y(\Delta y-u)p_Y(u)du$$results in the square root of a quartic in the denominator, which seems like a dead end.

I instead tried characteristic functions. From tables of Fourier transforms (FT's), I find that the CF of $p_Y$ is a Bessel function$$\varphi_Y(z)=\frac{1}{2\pi}J_0(z)$$$p_{\Delta Y}$ is then the inverse FT of the product of two of these CF's$$p_{\Delta Y}=F^{-1}(\varphi_{\Delta Y})=F^{-1}(\varphi_Y^2)=\frac{F^{-1}\left(J_0^2(z)\right)}{4\pi^2}$$but I cannot find the inverse FT of the square of the Bessel function. Can anyone help me finish this off?

marcusl said:
A direct approach involving the convolution$$p_{\Delta Y}(\Delta y)=p(y)*p(y)=\int_{-\infty}^{\infty}p_Y(\Delta y-u)p_Y(u)du$$...
That doesn't look quite right to me. I think it should be:
$$p_{\Delta Y}(\Delta y)=p(y)*p(y)=\int_{-1}^{1}p_Y(\Delta y+u)p_Y(u)du$$
which is equal to:
$$\int_{-1-\min(\Delta y,0)}^{1-\max(\Delta y, 0)}p_Y(\Delta y+u)p_Y(u)du$$
I would be inclined to stop there and just use numerical integration. Expressing the result as a Bessel function may look more compact, but will still involve numerical integration to evaluate the Gamma function inside the Bessel sum.

## 1. What is a Pdf of Difference of Random Variables?

A Pdf (Probability density function) of a difference of random variables is a statistical function that describes the probability distribution of the difference between two random variables. It can be used to calculate the likelihood of a particular difference occurring between the two variables.

## 2. How is the Pdf of Difference of Random Variables calculated?

The Pdf of Difference of Random Variables is calculated using the convolution of the Pdfs of the individual random variables. This involves taking the integral of the product of the two Pdfs over all possible values of the difference between the two variables.

## 3. What is the significance of the Pdf of Difference of Random Variables?

The Pdf of Difference of Random Variables is important in statistics as it allows us to understand and predict the characteristics of the difference between two random variables. It can be used to make inferences and draw conclusions about the underlying distributions of the variables.

## 4. Can the Pdf of Difference of Random Variables be used for any type of random variables?

Yes, the Pdf of Difference of Random Variables can be used for any type of random variables, including continuous, discrete, and mixed variables. However, the calculation method may vary depending on the type of variables involved.

## 5. How can the Pdf of Difference of Random Variables be applied in real-life situations?

The Pdf of Difference of Random Variables can be applied in various fields such as finance, engineering, and social sciences. It can be used to model and analyze the difference between two variables, which can help in decision-making, risk management, and understanding complex systems.

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