Question

Let Xi = (i =1,2,3) be independently and normally distributed random variable with mean of 4 as variance i. state the distribution of the following random variable

i) V = X1+X2+X3

Answer 1

The sum of normally distributed random variables is also a normally distributed random variable.

Given
`n` random variables with
`X_i\sim\mathrm{Normal}(\mu_i,\sigma_i^2)`, their sum is

`\displaystyle\sum_(i=1)^n X_i \sim \mathrm{Normal}\left(\sum_(i=1)^n \mu_i, \sum_(i=1)^n \sigma_i^2\right)`

i.e. normally distributed with mean and variance equal to the sums of the means and variances of the
`X_i`.

In this case, each of
`X_1,X_2,X_3` are normally distributed with
`\mu=4` and
`\sigma^2` = ... I'm not sure what you meant for the variance, so I'll keep it symbolic. Then

`V = X_1+X_2+X_3 \sim \mathrm{Normal}(12, 3\sigma^2)`