ECE2191 Probability Models in Engineering
Tutorial 10: Random Sums and Sequences
Second Semester 2021
1. Suppose the COVID-19 cases in Wakanda follow the normal distribution N(549, 24).
(a) Find the expected value and variance of the sample mean variable with sample size n = 16 and n = 36.
(b) Plot and compare the sample mean distributions of the two cases above; i.e., when n = 16 and n = 36.
2. At Footscray market, the weight of a one-pound bag of carrots can be modelled using the normal distribution N(1.18, 0.072). Similarly, the weight of a three-pound bag of carrots also follows the normal distribution N(3.22, 0.092). Minsoo goes shopping for carrots and he wishes to know what is the probability that the sum of three one-pound bags exceeds the weight of one three-pound bag?
3. An automobile battery manufacturer claims that its midgrade battery has a mean life of 50 months with a standard deviation of 6 months. Suppose the distribution of battery lives of this particular brand is approximately normal.
(a) On the assumption that the manufacturer’s claims are true, find the probability that a randomly selected battery of this type will last less than 48 months.
(b) On the same assumption, find the probability that the mean of a random sample of 36 such batteries will be less than 48 months.
4. A container ship carries 100 containers through the Suez Canal. Let Xi be the weight (in tonnes) of the ith container on the ship. Suppose that the Xi
’s are IID and approximately normal, and E[Xi
] = m = 170 and σXi = σ = 30. Find the probability that the total weight of the containers on the ship exceeds 18,000 tonnes.
5. A group of 25 cyclists bring 70 litres of water for their cross country riding. If an average cyclist drinks 2 litres of water for the entire trip with standard deviation of 0.8 litres, what is the probability that the group would run out of water before reaching the destination? Assume that the distribution of water consumed is normal.
6. In a communication system, each data packet consists of 1000 bits. Due to noise, each bit may be received in error with probability of 0.1. It is assumed that bit errors occur independently. Find the probability that there are more than 120 errors in a certain data packet. (Hint: You can assume that the sum of random Bernoulli variables can be approximated by a normal random variable.)
7. The number of customers entering a store on a given day is Poisson distributed with mean λ = 10. The amount of money spent by a customer is uniformly distributed over (0, 100). Find the mean and variance of the amount of money that the store takes in on a given day.
8. A new Coronavirus, COVID-20, is very deadly and spreads out very quickly. If an indi-vidual member of a population gets the virus, he or she shows symptoms at the beginning of the next day. After showing symptoms he or she will infect a random number of other individuals, and then die at midnight. Every person has the same probability distribution for infecting new individuals; however, the number of new infections caused by different individuals are independent of each other. The distribution of the number N of infections is given by Pr[N = i] = pi
, for i ≥ 0. Denote by m the expected number of new infections by any individual; i.e.,
Let Xn be the number of infected individuals at day n of the beginning of the pandemic, and Show that E[Yn+1|Xn] = Yn.
9. Due to a COVID-19 outbreak in Wonderland, two new testing centers have been opened to take tests from 10,000 individuals. Each person independent from the others chooses one of the two testing sites for getting tested with equal probability. Therefore, the number of people going to each center is a random number.
(a) What is the minimum number of testing kits that each center should have to ensure everyone can get tested?
(b) What is the minimum number of testing kits that each center should have to ensure everyone can get tested with probability 0.99?
10. Let N be a random variable taking values from {1, 2, 3, . . .}. Let Xi be a sequence of independent random variables which are also independent of N, where E[Xi
] = E[X], the same for all i. Show that
11. [Optional] Derive the variance of where the random variables Xi
’s and N are defined similar to those in Problem 10.