A source $S$ has source words $w_1, w_2, \ldots, w_n$, with probabilities $p_1 \geq p_2 \geq \ldots \geq p_n > 0$. Let $C$ be a binary Huffman code for $S$, and let $l$ be the length of the longest code word in $C$. Let $L(C)$ be the expected length of $C$.
Suppose you replace $w_n$ by two source words $w_a$ and $w_b$ with positive probabilities $p_a$ and $p_b$ respectively, where $p_a + p_b = p_n$. Call the new Huffman Code $C’$. What is $L(C’) – L(C)$?
So what I understand is that $L(C) = l_1p_1 + l_2p_2 + \ldots + l_np_n$ and $L(C’) = l_1p_1 + l_2p_2 + \ldots + (l_ap_a + l_bp_b)$ So if I were to find the difference $L(C’) – L(C)$ I am thinking that I would simply be left with $l_ap_a + l_bp_b - l_np_n$, but I have a feeling there is more to this than I presume.
Any ideas or suggestions would be greatly helpful!