SUKKUR IBA UNIVERSITY Probability & Statistics Fall 2020 Assignment No. 2 Class: BS-V (CS/SE) A Maximum Marks: 10 Note: Submit up to Jan 10, 2021 (11:59pm) Q1: We asked 6 students how many times they rebooted their computers last week. There were 4 Mac users and 2 PC users. The PC users rebooted 2 and 3 times. The Mac users rebooted 1, 2, 2 and 8 times. Let C be a Bernoulli random variable representing the type of computer of a randomly chosen student (Mac = 0, PC = 1). Let R be the number of times a randomly chosen student rebooted (so R takes values 1,2,3,8). (a) Create a joint probability table for C and R. Be sure to include the marginal probability mass functions. (b) Compute E(C) and E(R). (c) Determine the covariance of C and R and explain its significance for how C and R are related. (A one sentence explanation is all that’s called for.) Are R and C independent? (d) Independently choose a random Mac user and a random PC user. Let M be the number of reboots for the Mac user and W the number of reboots for the PC user. (i) Create a table of the joint probability distribution of M and W, including the marginal probability mass functions. (ii) Calculate P(W >M). (iii) What is the correlation between W and M? Q2: A machine that produces ball bearings has initially been set so that the true average diameter of the bearings it produces is 0.500 inch. A bearing is acceptable if its diameter is within 0.004 inch of this target value. Suppose, however, that the setting has changed during the course of production, so that the bearings have normally distributed diameters with mean value 0.499 inch and standard deviation 0.002 inch. What percentage of the bearings produced will not be acceptable? Q3: The time until a small meteorite first lands anywhere in the Sahara desert is modeled as an exponential random variable with a mean of 10 days. The time is currently midnight. What is the probability that a meteorite first lands sometime between 6 am and 6 pm of the first day? Q4: Test two integrated circuits one after the other. On each test, the possible outcomes are 𝑎 (accept) and 𝑟 (reject). Assume that all circuits are acceptable with probability 0.9 and that the outcomes of successive tests are independent. Count the number of acceptable circuits 𝑋 and count the number of successful tests 𝑌 before you observe the first reject. (If both tests are successful, let 𝑌 = 2.) a) Draw a tree diagram for the experiment and find the joint probability mass function 𝑓 𝑥, 𝑦 . b) Find the probability of the event 𝐴 that 𝑋, the number of acceptable circuits equals 𝑌, the number of tests before observing the first failure. c) Find the conditional probability 𝑃 𝑋 > 0 𝑌 = 0 . d) Find 𝜎𝑋𝑌 and 𝜌𝑋𝑌 . Q5: A company website has three pages. They require 750 kilobytes, 1500 kilobytes, and 2500 kilobytes for transmission. The transmission speed can be 5 Mb/s for external requests or 10 Mb/s for internal requests. Requests arrive randomly from inside and outside the company independently of page length, which is also random. The probability models for transmission speed, 𝑅, and page length, 𝐿, are: 0.4 𝑟 = 5, 𝑃𝑅 𝑟 = 0.6 𝑟 = 10, 0 𝑜. 𝑤. 0.3 𝑙 = 750, 𝑃𝐿 𝑙 = 0.5 𝑙 = 1500, 0.2 𝑙 = 2500, 0 𝑜. 𝑤. Write an expression for the transmission time 𝑔(𝑅, 𝐿) seconds. Derive the expected transmission time 𝐸 𝑔 𝑅, 𝐿 . Does 𝐸 𝑔 𝑅, 𝐿 = 𝑔 𝐸 𝑅 , 𝐸 𝐿 ? Q6: A modem transmits over an error-prone channel, so it repeats every “0” or “1” bit transmission five times. We call each such group of five bits a “codeword.” The channel changes an input bit to its complement with probability p = 1/10 and it does so independently of its treatment of other input bits. The modem receiver takes a majority vote of the five received bits to estimate the input signal. Find the probability that the receiver makes the wrong decision. Stay Safe