Sums of independent normal and of independent Poisson random variables

Consider two independent random variables \(\xi_1\) and \(\xi_2\) that are distributed according to the normal distribution with mean values \(a_1\), \(a_2\) and variances \(\sigma_1^2\), \(\sigma_2^2\) correspondingly. Find the distribution of a random variable \(\xi_1 + \xi_2\). Repeat the exercise in case if \(\xi_{1,2}\) are distributed according to the Poisson distribution with parameters \(\lambda_1\) and \(\lambda_2\).