Filter front page posts by category

Try adding this code to functions.php file:

add_action('pre_get_posts', 'ad_filter_categories');
function ad_filter_categories($query) {
    if ($query->is_main_query() && is_home()) {
        $query->set('category_name','news, uncategorized');
    }
}

category_name is the slug or the nicename of the category. Add a comma separated list of the categories you wish to include.