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

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