# HW4 (Due 12/5)

1. (5 points) Consider a binary channel with a cross over probability of 0.1 and an erasure probability of 0.1, what is the capacity of the channel? In other words, when a bit is sent through the channel, there is a probability of 0.1 that the bit is flipped and then there is an additional probability of 0.1 that the bit got erased (the decoder can't even tell if the received bit is 0 or 1).

2. (5 points) Consider a channel composed of two parallel Gaussian channels with noise powers $${N_1}=1$$ and $${N_2}=5$$. Given a total power $$P$$, you may assign power $$P_1$$ to channel 1 and power $$P_2$$ to channel 2 such that $$P_1+P_2=P$$. By trading off the power between the two channels, compute the (maximum) overall capacity: $$\frac{1}{2} \log(1+\frac{P_1}{N_1}) + \frac{1}{2} \log(1+\frac{P_2}{N_2}$$) for

1. $$P = 12$$

2. $$P = 4$$

3. (Extra credit-5 points) For the above question, find a general expression of the overall capacity for any $$P$$.

4. (5 points) For the following corpus containing the following sentences (each sentence considered as one document), 1. “Be not afraid of greatness. Some are born great, some achieve greatness, and others have greatness thrust upon them.” 2. “We know what we are, but know not what we may be.” 3. “Sweet are the uses of adversity which, like the toad, ugly and venomous, wears yet a precious jewel in his head.” 4. “Our doubts are traitors and make us lose the good we oft might win by fearing to attempt.” 5. “Give every man thy ear, but few thy voice.” 6. “Uneasy lies the head that wears the crown.” 7. “How poor are they that have not patience! What wound did ever heal but by degrees?” 8. “Nothing can come of nothing.”

1. Find the TF-IDF (see slide 55 here) for 'but’ in Document (Sentence) 2.

2. Find the TF-IDF for 'thy’ in Document (Sentence) 5.

5. (Extra credit-5 points) For the above question, compute the TF-IDF for each word in each documents and find the word with maximum the TF-IDF.

6. (5 points) For the simple Bayesian network below,

1. Identify a pair of independent variables

2. Identify a pair of conditionally independent variables given some variables