You can use this script to get category product collection randomly and you can exclude items which is not require..

<?php
    $objectManager = \Magento\Framework\App\ObjectManager::getInstance();
    $categoryInstance = $objectManager->create('Magento\Catalog\Model\CategoryFactory');
    $categoryId = 4;
    $excludeProductIds = array(2,3,4,5,6);
    $filteredProductCollection = $categoryInstance->create()->load($categoryId)->getProductCollection()->addAttributeToSelect('*')->setPageSize(15)->addFieldToFilter('entity_id', array('nin' => $excludeProductIds));

    $filteredProductCollection->getSelect()->orderRand();

    $filteredProductCollection->getItems();

    foreach ($filteredProductCollection as $product) {
        echo $product->getId()." => ".$product->getName();
        echo "<br/>";
    }
?>

Hope this will help you!

You can use this script to get category product collection randomly and you can exclude items which is not require..

<?php
    $objectManager = \Magento\Framework\App\ObjectManager::getInstance();
    $categoryInstance = $objectManager->create('Magento\Catalog\Model\CategoryFactory');
    $categoryId = 4;
    $excludeProductIds = array(2,3,4,5,6);
    $filteredProductCollection = $categoryInstance->create()->load($categoryId)->getProductCollection()->addAttributeToSelect('*')->setPageSize(15)->addFieldToFilter('entity_id', array('nin' => $excludeProductIds));

    $filteredProductCollection->getSelect()->orderRand();

    $filteredProductCollection->getItems();

    foreach ($filteredProductCollection as $product) {
        echo $product->getId()." => ".$product->getName();
        echo "<br/>";
    }
?>

You can use below code instead of using $objectManager directly..

<?php
    protected $_categoryFactory;

    public function __construct(
        \Magento\Catalog\Model\CategoryFactory $categoryFactory
    ) {
        $this->_categoryFactory = $categoryFactory;
    }

    public function execute()
    {
        $categoryId = 4;
        $excludeProductIds = array(2,3,4,5,6);
        $filteredProductCollection = $this->_categoryFactory->create()->load($categoryId)->getProductCollection()->addAttributeToSelect('*')->setPageSize(15)->addFieldToFilter('entity_id', array('nin' => $excludeProductIds));

        $filteredProductCollection->getSelect()->orderRand();

        $filteredProductCollection->getItems();

        foreach ($filteredProductCollection as $product) {
            echo $product->getId()." => ".$product->getName();
            echo "<br/>";
        }
    }
?>

Hope this will help you!