Reference no: EM132378757

CSC 601 Advanced Computing Science and Applications Assignment - King Saud University, Saudi Arabia

1. We are given a number of darts. When we throw a dart at a target, we have a probability of 1/4 of hitting the target. What is the probability of obtaining at least one hit if three darts are thrown? Calculate this probability in two ways.

Hint Construct the sample space. How many outcomes are in the sample space? Are all outcomes in the sample space equally likely?

2. Manufacturer X produces personal computers (PCs) at two different locations in the world. Fifteen percent of the PCs produced at location A are delivered defective to a retail outlet, while 5 percent of the PCs produced at location B are delivered defective to the same retail store. If the manufacturing plant at A produces 1,000,000 PCs per year and the plant at B produces 150,000 PCs per year, find the probability of purchasing a defective PC.

3. Suppose the arrival of telephone calls at a switch can be modeled with a Poisson PMF. That is, if X is the number of calls that arrives in t minutes, then: P_r (X = k) = (λ_t) k k! e-λt, k = 0, 1, 2, · · · where λ is the average arrival rate in calls/minute. Suppose that the average rate of calls is 10 per minute.

(a) What is the probability that fewer than three calls will be received in the first 6 seconds?

(b) What is the probability that fewer than three calls will be received in the first 6 minutes?

4. In a digital communication system, a block of k data bits is mapped into an n bit codeword that typically contains the k information bits as well as n - k redundant bits. This is known as an (n, k) block code. The redundant bits are included to provide error correction capability. Suppose that each transmitted bit in our digital communication system is received in error with probability p. Furthermore, assume that the decoder is capable of correcting any pattern of t or fewer errors in an n bit block. That is, if t or fewer bits in an n bit block are received in error, then the codeword will be decoded correctly, whereas if more than t errors occur, the decoder will decode the received word incorrectly. Assuming each bit is received in error with probability p = 0.03, find the probability of decoder error for each of the following codes.

(a) (n, k) = (7, 4), t = 1;

(b) (n, k) = (15, 7), t = 2;

(c) (n, k) = (31, 16), t = 3;

(d) Comment on your results (any trends? are they expected?).

5. A certain random variable has a probability density function of the form

fX(x) = ce^-2xu(x). Find the following:

(a) the constant c;

(b) mean and variance;

(c) cdf, and then:

i. Pr(X > 2);

ii. Pr(X < 3);

iii. Pr(X < 3/X > 2).

6. Determine the value of c that makes the function fXY (x, y) = ce-2x-3y a joint probability density function over: x > 0 and y > 0. Determine the following: (a) P(X < 1, Y < 2); (b) P(Y > 3); (c) E(X); (d) E(Y); (e) E(XY).

7. Trains A and B arrive on two separate tracks at a station at random between 8 AM and 8:10 AM. Train A stops for 2 minutes and train B stops for 3 minutes. Assuming that the trains arrive independently of each other, what it the probability that the two trains will meet at the station (i.e., there is an overlap in their stopping periods)?