Question

Let X1 be a normal random variable with mean µ1 and variance σ 2 1 , and let X2 be a normal random variable with mean µ2 and variance σ 2 2 . Assuming that X1 and X2 are independent, what is the distribution of X1 + X2? g

Answer 1

`X_1 + X_2 \sim (\mu_1 + \mu_2, \sigma_1^2 + \sigma_2^2 )`

Explanation:

We are given the following in the question:

`X_1` is a random normal variable with mean and variance

`\mu_1\\\sigma_1^2`

`X_1 \sim N(\mu_1,\sigma_1^2)`

`X_2` is a random normal variable with mean and variance

`\mu_2\\\sigma_2^2`

`X_2 \sim N(\mu_2,\sigma_2^2)`

`X_1, X_2` are independent events.

Let

`Z =X_1 + X_2`

Then, Z will have a normal distribution with mean equal to the sum of the two means and its variance equal the sum of the two variances.

Thus, we can write:

`\mu = \mu_1 + \mu_2\\\sigma^2 = \sigma_1^2 + \sigma_2^2\\Z \sim (\mu_1 + \mu_2, \sigma_1^2 + \sigma_2^2 )`