G) For two random variables 𝑋 and 𝑌, 𝐸(𝑋𝑌) = 𝐸(𝑋)𝐸(𝑌) hold if 𝑋 and 𝑌 are _______

b) Find 𝑝𝑌 (𝑦), the marginal p.m.f. of 𝑌

Question 2(c)Let 𝑋1 → 𝑋2 → 𝑋3 → · · · → 𝑋𝑛 form a Markov chain in this order. Thus, the joint probabilityof 𝑋1, . . . , 𝑋𝑛 are given by𝑝(𝑥1, 𝑥 − 2, . . . , 𝑥𝑛) = 𝑝(𝑥𝑛 |𝑥𝑛−1) 𝑝(𝑥𝑛−1 |𝑥𝑛−2) · · · 𝑝(𝑥2 |𝑥1) 𝑝(𝑥1).1. Express 𝐼 (𝑋1; 𝑋2, . . . , 𝑋𝑛) in terms of its entropy and conditional entropy [2 Marks]2. Simplify the entropy expression you derived above to reduce 𝐼 (𝑋1; 𝑋2, . . . , 𝑋𝑛) to itssimplest form. Please do not use other methods to simplify, you will not receive anymarks.

eet the joint p.d.f. of X1 and X2 be:ℎ(𝑥1, 𝑥2) = {8𝑥1𝑥2 for 0 < 𝑥1 < 𝑥2 < 10 otherwisea) Find the joint p.d.f. of 𝑌1 = 𝑋1𝑋2and 𝑌2 = 𝑋2b) Are 𝑌1 and 𝑌2 independent? Why? (10 points

If lim𝑥→2𝑓(𝑥)=3𝑥 and lim𝑥→2𝑔(𝑥)=4𝑥2−5, what is lim𝑥→2𝑔(𝑓(𝑥))?

(0,1)3. Given the relationship𝑦 = 𝑎* + 𝑎&𝑥& + 𝑎!𝑥! + 𝑎+𝑥+Where 𝑥&, 𝑥!, 𝑥+ are model variables; and 𝑎*, 𝑎&, 𝑎!, 𝑎+ are constants; Showhow the errors in 𝑥&, 𝑥!, 𝑥+ affect the value of 𝑦 if they are:i. Blundersii. Systematiciii. Randomiv. Random and uncorrelated2. A