1 Question # 1 A digital transmission carries a randomly mixed set of voice packets. The occurrence statistics of packets are designated in 6 groups. The number of occurrences of each group and the corresponding random value of the associated bit rates are as follows: Event enumerated Bit rate (Ri) Voice in kbps Packets Number of occurrences (ni) i=1 64 i=2 59 i=3 60 i=4 62 i=5 57 i=6 61 65 62 55 58 50 60 Determine the following of the RV (Bit rate - Ri): 1. Find the upper and lower bounds of the average performance of the RV 2. State with reason the most appropriate mean value that describes the average performance of the bit rate 3. Find the expected mean, median and mode --------------------------------------------------------------------------------------------------------------------- 2 Question # 2 A digitized video transmission carries a randomly mixed set of video packets. The occurrence statistics of the packets are designated in 6 groups. The video transmission involves streaming of packets wherein the statistics of occurrence of bit rate in each packet depends strongly on bit-rate values in consequent packets streamed. The number of occurrences of each group and the corresponding random value of the associated bit rates are as follows: Event enumerated Bit rate (Ri) Video in Mbps Packets Number of occurrences (ni) i=1 6.4 i=2 5.9 i=3 0.6 i=4 6.2 i=5 5.7 i=6 6.1 85 60 50 25 50 60 Determine the following of the RV (Bit rate - Ri): 1. Find the upper and lower bounds of the average performance of the RV 2. State with reason the most appropriate mean value that describes the average performance of the bit rate 3. Find the expected mean, median and mode --------------------------------------------------------------------------------------------------------------------- 3 Question # 3 (A) The probability distribution function of a continuous variable is as follows: p(x) = (1/2)sin(x) 0x =0 else where (i) (ii) Find the expected mean Determine the cumulative probability between 0 to /2 (Useful integration formula: [x sin(x)]dx = sin(x) - [x cos(x)] (B) Indicated below is a set of eight numbers x assigned to directions (N, S, E, W, NE, NW, SE and SW) of a hurricane-track respectively; and, the corresponding occurrence probabilities of the directions are, p(x) as listed. x p(x) 2 0.1 3 0.15 4 5 0.25 0.1 6 0.05 7 8 9 0.05 0.15 0.15 Determine: (i) Expected mean of x; (ii) Standard deviation of the distribution; (iii) Decide the most probable direction based on the central tendency performance (iv) Plot: CDF of x (C) A set of computer product manufactured is tested. Suppose x denotes the number of units passing the test, find the standard deviation of x. x in 1000s p(x) 0 1/8 1 2 3 3/8 3/8 1/8 4 Question # 4 (A) The life-time of a perishable product specified in months is modeled by a random variable (RV), x with a pdf given by: p(x) = k(9 - x) 0 for 0 x 9 otherwise where k is a positive constant. Find: (i) (ii) Value of k; and, Expected value of the life-time. (B) Two identical routers are located at site A with 8 ports and at site B, with 10 ports. In how many ways can these be connected to 5 access points placed on the network at the same time? (C) A Cisco edge-router in a WAN receives requests for connectivity on an average of 6 per minute. Assume each request is independent of other requests. Determine the probability of getting exactly 8 requests in one minute. 5 Question # 5 (A) In using a mission-critical computer operation, the probability of the state: \"No software failure\" is denoted as p(1); and, the probability of corresponding failure state is indicated as p(0). Suppose p(1) = 0.6. What is the chance of 10 computers will successfully function, if 12 computers are deployed in the mission? (B) A uniformly distributed RV, x is specified within the range, 0 to 100. It is sampled 10 times at equal intervals yielding 10 discrete samples. Find the variance of the continuous random variable, x (i) Estimate the variance of the 10 sampled values. (ii) Is there a difference between the variances calculated in (a) and (b)? If so, why? ------------------------------------------------------------------------------------------------------------ 6 Question # 6 A If a Poisson distributed RV, x is such that its probability at (x = 0) is equal to the probability at: (x = 1) plus 3 probability at (x = 2). Find the mean of x. B. In a statistically large number of enumerated wireless connectivity, x depicts the statistics of call failures with an expected mean, o = 50. Assume that x shows a strong central tendency towards its mean value; and, the corresponding standard deviation, o = 14. Suppose, the calls are randomly monitored and the sampled observation shows 49 call failures. Mostly xL = 30 such failures are on the lower side and xU = 70 failures on the upper side of the sampled mean value. Construct the normalized distribution of Z, p (Z) versus Z showing ZsL; and ZsU