Question

Cauchy distribution

Also known as the Lorentz distribution, especially among physicists, and named after French mathematician Augustin-Louis Cauchy, (although it had been studied before, in particular by Poisson) the Cauchy distribution is a family of continuous probability distributions. It has been used in many applications such as mechanical and electrical theory, physical anthropology, measurement problems, risk and financial analysis. It was also used to model the points of impact of a fixed straight line of particles emitted from a point source. The density function fₓ(x) of a (standard) Cauchy distribution is defined as:

fₓ(x)=π(1+x²)/1, for x∈R.
(a) Determine the cumulative distribution function Fₓ(x), and then find the inverse function F ₓ⁻¹of FX[3 marks]
(b) Describe an algorithm which uses the inverse transform method to generate n values sampled from a standard Cauchy distribution. [ 2 marks]
(c) Apply your methodology to generate a sample of 100000 values from a Cauchy distribution. Plot a normalised histogram (like a probability density) of these realisations. [ 2 marks]
(d) On that same histogram, add a line that shows the probability density fuction of the standard Cauchy distribution. [ 2 marks]
(e) By using only the U(0,1) distribution as a source of randomness, find a way to calculate the following integral with R : ∫⁷₅fₓ(x)dx [Hint: first, a suitable (linear) change of variable could be useful.] You can check your answer by hand or by using R functions. [3 marks]

Answer 1

The Cauchy distribution is a continuous probability distribution. The cumulative distribution function (CDF) and its inverse function can be determined using the density function of the Cauchy distribution. The inverse transform method can be used to generate values from a Cauchy distribution.

Step-by-step explanation:

The Cauchy distribution, also known as the Lorentz distribution, is a continuous probability distribution. The density function of a (standard) Cauchy distribution is defined as f(x) = π(1 + x²)/1, for x∈R. To determine the cumulative distribution function (CDF) F(x), we integrate the density function from negative infinity to x. Taking the inverse of the CDF, we can find the inverse function F⁻¹(x).

To generate n values sampled from a standard Cauchy distribution using the inverse transform method, we generate n values uniformly distributed between 0 and 1, and then use the inverse function F⁻¹(x) to obtain values from the Cauchy distribution.

To generate a sample of 100,000 values from a Cauchy distribution, we use the method described above, generate 100,000 values using the inverse transform method, and plot a normalized histogram of these realizations. We can then add a line to the histogram to show the probability density function of the standard Cauchy distribution.

To calculate the integral ∫⁷₅f(x)dx using only the U(0,1) distribution as a source of randomness, we can use the property of the Cauchy distribution that any linear transformation of a standard Cauchy random variable is also Cauchy distributed. We can then use a change of variable to transform the integral into the integral of the standard Cauchy distribution, which can be calculated using R functions or by hand.