2S2122 Activity 2.1

Instructions

• Answer the following problems individually and truthfully.
• Be sure to show your solutions and please box your final answers.
• Please write your complete name, student number, and section on the upper left corner of your answer sheet. No name, student number, and section, no grade.
• Save your answers in pdf file type with the filename format "section_lastname_firstname_studentnumber.pdf" all in small caps. For example: "abc_wayne_bruce_201101474.pdf".
• Submit your files in the respective submission bin in UVLE. Be sure to submit in the correct class!
• Have fun doing these exercises :) even though it may seem boring.
• You only have two weeks to work on the activities. We will post the hard deadline on UVLE.
• You might want to create your own programs that calculate information and entropy.

Grading Rubrics

• If you have a good solution and a correct answer, you get full points.
• If you have a good solution but the answer is not boxed (or highlighted), you get a 5% deduction of the total points for that problem.
• If you have a good solution but the answer is wrong, you get a 20% deduction of the total points for that problem.
• If your solution is somewhat OK but incomplete. You only get 40% of the total problem.
• If you have a bad solution but with a correct answer. That sounds suspicious. You get 0% for that problem. A bad solution may be:
  • You just wrote the given.
  • You just dumped equations but no explanation to where they are used.
  • You attempted to put a messy flow to distract us. That's definitely bad.
• No attempt at all means no points at all.
• Some problems here may be too long to write. You are allowed to write the general equation instead but be sure to indicate what it means.
• Make sure that you use bits as our units of information.

Problem 1 (1 pts.)

A word in a code consists of five binary digits (e.g., $10110$ is one code word). Each digit is chosen independently of the others and the probability of any particular digit being a 1 is $p=0.6$. Find the information associated with the following events:

(a) At least three 1s. (0.2 pts.)

(b) At most four 1s. (0.2 pts.)

(c) Exactly two 0s. (0.2 pts.)

(d) Let $X$ be a random variable representing the sum of 1s for the code with five binary digits. Calculate $H(X)$. (0.2 pts.)

(e) Interpret your results in (d) to (a), (b), and (c). Philosophize what you think of it. (0.2 pts.)

Problem 2 (2 pts.)

The popular table top game Dungeons and Dragons uses 6 types of dies. We denote each die as ${\textrm {D}}n$ where $n$ is an $n$-sided die. For example, a ${\textrm {D}}4$ is a 4-sided die. Each face of a die is fair (i.e., none of the faces are biased). Calculate and answer the following:

(a) $H({\textrm {D}}4)$ (0.2 pts.)

(b) $H({\textrm {D}}6)$ (0.2 pts.)

(c) $H({\textrm {D}}8)$ (0.2 pts.)

(d) $H({\textrm {D}}10)$ (0.2 pts.)

(e) $H({\textrm {D}}12)$ (0.2 pts.)

(f) $H({\textrm {D}}20)$ (0.2 pts.)

(g) Comment or interpret how does information vary per die? Is it consistent with your belief of uncertainty? (0.4 pts.)

(h) A potion of greater healing heals a character with the formula $4{\textrm {D}}4+4$. That means we need to roll a ${\textrm {D}}4$ die four times and add a $4$ to the sum of the rolls. How surprising is it to get $+20\ {\textrm {HP}}$. HP stands for hit-points or the life of a character. (Hint: Careful! 🙂) (0.4 pts.)

(i) (Bonus) Which is better? To perceive events as a chance or an uncertainty? If we like your answer you get +1 for the entire exercise.

Problem 3 (2 pts.)

$X$ and $Y$ are Bernoulli (or binary) random variables with the distribution of $X$ having $p={\frac {1}{3}}$ (i.e., $X$ has a ${\frac {1}{3}}$ chance of being a 1). We are also given the conditional probabilities:

• $P(Y=0|X=0)={\frac {1}{4}}$
• $P(Y=1|X=1)={\frac {3}{5}}$

Calculate the following:

(a) $H(X,Y)$ (0.4 pts.)

(b) $H(Y|X)$ (0.4 pts.)

(c) $H(Y)$ (0.4 pts.)

(d) $H(X|Y)$ (0.4 pts.)

(e) $I(X,Y)$ (0.4 pts.)

Problem 4 (2 pts.)

We have three urns $A$, $B$, and $C$, which contain colored balls as follows:

• $A$ has three red and five green balls.
• $B$ has one red and two green balls.
• $C$ has seven red and six green balls.

Let $X$ be the random variable representing the three urns $\{A,B,C\}$. Let $Y$ be the random variable representing $\{R,G\}$. Where $R$ represents "a red ball was chosen" and $G$ represents "a green ball was chosen". Determine the following:

(a) $H(X,Y)$ (1/3 pts.)

(b) $H(Y)$ (1/3 pts.)

(c) $H(X)$ (1/3 pts.)

(d) $H(Y|X)$ (1/3 pts.)

(e) $H(X|Y)$ (1/3 pts.)

(f) $I(X,Y)$ (1/3 pts.)

Problem 5 (3 pts.)

Process writing is where we re-write or re-derive concepts that were taught to us. It is another form of improving our understanding of a concept: "reteaching what was taught ... " . For this problem, rederive the bounds of entropy. Suppose we are given some random variable $X$ we were taught that $0\leq H(X)\leq \log _{2}(n)$. Show and explain in your own words how:

• $H(X)\geq 0$ (0.5 pts.)
• $H(X)\leq \log _{2}(n)$ (2.5 pts.)