1. The following table provides the joint probabilities P(X=ai, Y=bj) of the values that two random variables X and Y take (X takes values 2,4,6,8,10,12 and Y takes values 0,2,5,7).
X 2 4 6 8 10 12
0 0 0.01 0.02 0.03 0.04 0.1
2 0.01 0.01 0.05 0.1 0.09 0.04
5 0.04 0.09 0.1 0.05 0.01 0.01
7 0.1 0.04 0.03 0.02 0.01 0
Find the means, variances, and standard deviations of X and Y and the
correlation between X and Y.
2. In problem 1, find a regression for estimating Y from X and determine the variance of the error. What is the estimated value of Y when X=5?
3. We collected data during 20 seconds in a network, where n=900 packets were sent and 108 of them were lost. Let p denote the probability that a packet is lost. Estimate the value of p and provide an error margin at a confidence level C=95%.
4. What would be the estimate in problem 3, if we wanted a one-sided error margin on the estimate at the same confidence level 95%.
5. In problem 4, what should be the number of packets n used in the sample (instead of 900) so that we can state that p is less than 0.14 with 99% confidence level?
6. For the same data collected in problem 3, determine a test that verifies the hypothesis that the probability of a lost packet is larger than 0.1 at significance level ? =0.025. Does the data provided in problem 3 supports this hypothesis? If yes, find the P-value of the test.
7. If the true value of p = 0.14. What is the probability of Type II error in the test you designed in problem 6?
8. Let p as obtained from problem 3 be the true value of loss probability of a packet. Suppose we send 9 packets in 0.2 seconds, what is the probability that we lose more than one packet?
9. Consider a pure ALOHA system with a 56 kbits/sec channel with N users.
Each user sends one message on average every 100 seconds. The length of a message is fixed at 1000 bits (including overhead). What is the maximum number of users N?
10. Consider a slotted ALOHA system with a slot size of 0.02 seconds and
message size of 512 bits. In a lightly loaded condition the load (including original and retransmitted messages) is G = 0.6 messages per slot. Find the probability of a collision, the throughput in both messages per slot and k-bits per second.
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