 Mathematical Foundations of Computer Networking: Probability

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This chapter is from the book

1.10. Exercises

1. Sample space

In the IEEE 802.11 protocol, the congestion window (CW) parameter is used as follows: Initially, a terminal waits for a random time period, or backoff, chosen in the range [1, 2CW] before sending a packet. If an acknowledgment for the packet is not received in time, CW is doubled, and the process is repeated until CW reaches the value CWMAX. The initial value of CW is CWMIN. What are the sample spaces for the value of CW and the value of the backoff?

2. Interpretations of probability

Consider the statement: Given the conditions right now, the probability of a snowstorm tomorrow morning is 25%. How would you interpret this statement from the perspective of an objective, frequentist, and subjective interpretation of probability, assuming that these are possible?

3. Conditional probability

Consider a device that samples packets on a link.

• a. Suppose that measurements show that 20% of packets are UDP and that 10% of all packets are UDP packets with a packet size of 100 bytes. What is the conditional probability that a UDP packet has size 100 bytes?

b. Suppose that 50% of packets were UDP, and 50% of UDP packets were 100 bytes long. What fraction of all packets are 100-byte UDP packets?

4. Conditional probability again

Continuing with Exercise 3: How does the knowledge of the protocol type change the sample space of possible packet lengths? In other words, what is the sample space before and after you know the protocol type of a packet?

5. Bayes’s rule

For Exercise 3(a), what additional information do you need to compute P(UDP|100)? Setting that value to x, express P(UDP|100) in terms of x.

6. Cumulative distribution function (CDF)

1. Suppose that discrete random variable D take values {1, 2, 3,...,i,...} with probability 1/2i. What is its CDF?
2. b. Suppose continuous random variable C is uniform in the range [x1, x2]. What is its CDF?
7. Expectations

Compute the expectations of the D and C in Exercise 6.

8. Variance

Prove that V[aX] = a2V[X].

9. Moments

Prove that .

10. MGFs

Prove that the MGF of a uniform random variable, expressed in terms of its series expansion, is .

11. MGFs

Prove that the rth moment of the uniform distribution about the origin is 1/(r+1).

12. MGF of a sum of two variables

Use MGFs to find the variance of the sum of two independent uniform standard random variables.

13. MGF of a normal distribution

Prove that if X ~ N(μ,σ2), then (X –μ)/σ ~ N(0,1).

14. Bernoulli distribution

A hotel has 20 guest rooms. Assuming that outgoing calls are independent and that a guest room makes 10 minutes worth of outgoing calls during the busiest hour of the day, what is the probability that 5 calls are simultaneously active during the busiest hour? What is the probability of 15 simultaneous calls?

15. Geometric distribution

Consider a link that has a packet loss rate of 10%. Suppose that every packet transmission has to be acknowledged. Compute the expected number of data transmissions for a successful packet+ack transfer.

16. Poisson distribution

Consider a binomially distributed random variable X with parameters n = 10, p = 0.1.

1. Compute the value of P(X = 8), using both the binomial distribution and the Poisson approximation.
2. Repeat for n = 100, p = 0.1.
17. Gaussian distribution

Prove that if X is Gaussian with parameters (μ, σ2), the random variable Y = aX + b, where a and b are constants, is also Gaussian, with parameters (aμ + b, (aσ)2).

18. Exponential distribution

Suppose that customers arrive at a bank with an exponentially distributed interarrival time with mean 5 minutes. A customer walks into the bank at 3 p.m. What is the probability that the next customer arrives no sooner than 3:15?

19. Exponential distribution

It is late August and you are watching the Perseid meteor shower. You are told that the time between meteors is exponentially distributed with a mean of 200 seconds. At 10:05 p.m., you see a meteor, after which you head to the kitchen for a bowl of ice cream, returning outside at 10:08 p.m. How long do you expect to wait to see the next meteor?

20. Power law

Consider a power-law distribution with xmin = 1 and α = 2 and an exponential distribution with λ = 2. Fill in the following table:

 x fpower_law(x) fexponential(x) 1 5 10 50 100

It should now be obvious why a power-law distribution is called heavy-tailed!

21. Markov’s inequality

Consider a random variable X that exponentially distributed with parameter λ = 2. What is the probability that X > 10 using (a) the exponential distribution and (b) Markov’s inequality?

22. Joint probability distribution

Consider the following probability mass function defined jointly over the random variables X, Y, and Z:

• p(000) = 0.05; p(001) = 0.05; p(010) = 0.1; p(011) = 0.3;
• p(100) = 0.05; p(101) = 0.05; p(110) = 0.1; p(111) = 0.3.
1. Write down pX, pY, pZ, pXY, pXZ, pYZ.
2. Are X and Y, X and Z, or Y and Z independent?
3. What is the probability that X = 0 given that Z = 1.