HW2S24_Solutions (1)

EE250: Probability, Random Variables and Stochastic Processes Spring 2024 Homework 2 Solutions Due 11:59PM, Feb. 27 th 2024 Notes: a. Each submission must be a single pdf file. b. Solutions should be submitted on Canvas. c. This assignment is worth 100 points. 1. Monitor three consecutive packets going through an Internet router. Classify each one as either video (v) or data (d). Your observation is a sequence of three letters (each one is either v or d). For example, three video packets corresponds to vvv. The outcomes vvv and ddd each have probability 0.2 whereas each of the other outcomes vvd, vdv, vdd, dvv, dvd, and ddv has probability 0.1. Count the number of video packets Nv in the three packets you have observed. Calculate the following probabilities: (a) P[Nv=2] (b) P[Nv≥1] (c) P[{vdv}|Nv=2] (d) P[{ddd}|Nv=1] (e) P[Nv=2| Nv≥1] (f) P[Nv≥1| Nv=2]

2. A ternary communication system is shown in Figure below. Suppose that input symbols 0, 1, and 2 occur with probability 1/3 respectively. (a) Find the probabilities of the output symbols, i.e., P[the output symbol=i], i=0,1,2 (b) Suppose that a 1 is observed at the output.What is the probability that the input was 0? 1? 2?

(b) Find the probability that coin 1 was tossed given that 3 heads were observed. (a) Event A: the number of heads is 3 Event B1: the first coin is selected Event B2: the second coin is selected P[A]=P[A|B1]P[B1]+ P[A|B2]P[B2]= ³ 2 ² ´ ² 1 2 µ ³ 1 ² ´ ² 1 2 (b) P[B1|A]= ±[h1¶·] ±[·] = ± · h1 ±[h1] ±[·] = ³ 2 ² ´ ² 1 2 ³ 2 ² ´ ² 1 2 µ³ 1 ² ´ ² 1 2 4. A lot of 15 items has 10 blue items and 5 red items (a) Suppose we select 7 items from the lot, with replacement, with order. Let event C be: there are 4 blue items and 3 red items in the sample. Find P[C]. (b) Suppose we select 7 items from the lot, with replacement, without order. Let event D be: there are 4 blue items and 3 red items in the sample. Find P[D]. (a) There are 7 15 outcomes (each is a sequence) in the sample space. There are       4 7 5 10 3 4 outcomes in the event C. (the number of orders for the 4-blue items (blue order) is 4 10 , the number of orders for the 3-red items (red order) is ¸ ² , since each sequence in event C includes 7 items, ¹ º is the number of ways to mix the blue order and red order) P[C]= 7 3 4 15 4 7 5 10       (b) There are         7 7 1 15 outcomes in the sample space. There are                 3 3 1 5 4 4 1 10 outcomes in the event D. (the number of 4-blue items is         4 4 1 10 , the number of 3-red items is         3 3 1 5

                         7 7 1 15 3 3 1 5 4 4 1 10 ] [ D P 5.