When working with random variables, it is often necessary to calculate the distribution of their sum. In Julia, there are several ways to solve this problem. In this article, we will explore three different approaches and determine which one is the most efficient.

## Approach 1: Using the Convolution Operator

One way to calculate the distribution of the sum of random variables is by using the convolution operator. The convolution of two probability distributions is defined as the distribution of the sum of two independent random variables, each with one of the original distributions.

```
using Distributions
# Define the probability distributions
dist1 = Normal(0, 1)
dist2 = Normal(0, 2)
# Calculate the distribution of the sum
dist_sum = dist1 + dist2
```

In this approach, we use the `Normal`

distribution from the `Distributions`

package to define the probability distributions. We then use the `+`

operator to calculate the distribution of the sum.

## Approach 2: Sampling and Summing

Another approach is to sample from each probability distribution and sum the samples to obtain an empirical distribution of the sum. This method is useful when the probability distributions are not analytically tractable.

```
using Random
# Set the seed for reproducibility
Random.seed!(123)
# Sample from the probability distributions
samples1 = rand(dist1, 1000)
samples2 = rand(dist2, 1000)
# Calculate the distribution of the sum
sum_samples = samples1 + samples2
```

In this approach, we use the `rand`

function from the `Random`

module to sample from each probability distribution. We then sum the samples to obtain an empirical distribution of the sum.

## Approach 3: Using the Characteristic Function

The characteristic function of a random variable is the Fourier transform of its probability distribution. By taking the product of the characteristic functions of two random variables, we can obtain the characteristic function of their sum. Inverse Fourier transforming the characteristic function gives us the distribution of the sum.

```
using FFTW
# Define the characteristic functions
cf1(t) = exp(-0.5im*t)
cf2(t) = exp(-2im*t)
# Calculate the characteristic function of the sum
cf_sum(t) = cf1(t) * cf2(t)
# Calculate the distribution of the sum
dist_sum = ifft(cf_sum)
```

In this approach, we define the characteristic functions of the probability distributions. We then multiply the characteristic functions to obtain the characteristic function of the sum. Finally, we use the inverse Fourier transform to obtain the distribution of the sum.

After comparing the three approaches, it is clear that the first approach using the convolution operator is the most efficient. It provides an analytical solution without the need for sampling or Fourier transforms. However, the choice of approach depends on the specific problem and the availability of analytical solutions for the probability distributions involved.