Reference no: EM132590100

EE524 Assignment -

Assume the valid parameters wherever it is necessary.

Q1. In any office, 12 telegrams come in a week. The telegrams are randomly distributed among all the days. What is the probability that at least one telegram comes each day?

Q2. Two players, A and B, alternately and independently flip a coin and the first player to obtain a head wins. Assume player A flips first.

(a) If the coin is fair, what is the probability that A wins?

(b) Suppose that Pr(head) = p, not necessarily ½. What is the probability that A wins?

(c) Show that for all p, 0 < p < 1, Pr(A wins) > ½.

Q3. The next generation of miniaturized wireless capsules with active locomotion will require two miniature electric motors to maneuver each capsule. Suppose 10 motors have been fabricated but that, in spite of tests performed on the individual motors, 2 will not operate satisfactorily when placed into a capsule.

To fabricate a new capsule, 2 motors will be randomly selected.find the probability that

(a) at least one motor will operate satisfactorily in the capsule

(b) at most one motor will operate satisfactorily

Q4. Seven balls are distributed randomly into seven cells. Let X_i = the number of cells containing exactly I balls. What is the probability distribution of X₃?

Q5. Among 60 automobile repair parts loaded on a truck in San Francisco, 45 are destined for Seattle and 15 for Vancouver. If two of the parts are unloaded in Portland by mistake and the "selection" is random, what are the probability that

(a) both parts should have gone to Seattle;

(b) both parts should have gone to Vancouver;

(c) one should have gone to Seattle and one to Vancouver?

Q6. A computer will be damaged with probabilities 0.01, 0.001, and 0.05, respectively, if its power supply has a voltage X below 100V, in between 100V and 120V, and above 120V. Suppose that X ~ N(110V,(10V)²). Find

(a) the probability that the computer will be damaged

(b) the probability that the voltage is above 120V if the computer is damaged.

Q7. Let B₁, B₂, . . . , B_n be a set of mutually exclusive and exhaustive events. Then proof that,

Pr(A) = _i=1∑^mPr(A|B_i)Pr(B_i).

Q8. Engineers in charge of maintaining our nuclear fleet must continually check for corrosion inside the pipes that are part of the cooling systems. The inside condition of the pipes cannot be observed directly but a non-destructive test can give an indication of possible corrosion. This test is not infallible. The test has probability 0.7 of detecting corrosion when it is present but it also has probability 0.2 of falsely indicating internal corrosion. Suppose the probability that any section of the pipe has internal corrosion is 0.1.

(a) Determine the probability that a section of pipe has internal corrosion, given that the test indicates its presence.

(b) Determine the probability that a section of pipe has internal corrosion, given that the test is negative.

Q9. Consider X follows binomial distribution, i.e X ~ B(n, p). If the number of trial (n) is large, and the probability of success (p) is small. Then prove that

Pr(X=k) = e^-λλ^k/k!.

Q10. Two movie theaters compete for the business of 1000 customers. Assume that each customer chooses between the movie theaters independently and with "indifference". Let N denote the number of seats in each theater.

(a) Using a binomial model, find an expression for N that will guarantee that the probability of turning away a customer is less than 1%.

(b) Use the normal approximation to get a numerical value for N.

Q11. Suppose X has a binomial (n, p) distribution and let Y has a negative binomial (r, p) distribution. Show that F_X(r-1) = 1 - F_Y(n-r).

Q12. When a relay tower for wireless phone service breaks down, it quickly becomes an expensive proposition for the phone company, and the cost increases with the time it is inoperable. From company records, it is postulated that the probability is 0.90 that the breakdown can be repaired within one hour. For the next three breakdowns, on different days and different towers,

(a) List all possible outcomes in terms of success, S, repaired within one hour, and failure, F, not repaired within one hour.

(b) Find the probability distribution of the number of successes, X, among the 3 repairs.

Q13. If a random variable X follows Poisson distribution i.e. Pr(X = k) = e^-λλ^k/k!, k = 0, 1, 2, . . .

(a) prove that _k=0∑^∞Pr(X=k) = 1,

(b) Calculate mean and variance of X.

Q14. As an example of a waiting-for-occurrence application, consider a telephone operator who, on the average, handles five cells every 3 minutes. What is the probability that there will be no calls in the next minute? At least two calls in the next minute?

Q15. Assuming that the typing mistakes per page committed by a typist follows a Poisson distribution, find the theoretical frequencies for the following distribution of typing mistakes.

Q16. Around one million people in the US (population around 308 million) have a certain particularly nasty condition, condition X. A test exists for X, which is 95% accurate. Being a hypochon-driac, I have myself tested for X, and the test comes back positive. What is the probability that I have X?

Q17. A truncated discrete distribution is one in which a particular class cannot be observed and is eliminated from the sample space. In particular, if X has range 0, 1, 2, . . . and the 0 class cannot be observed (as usually the case), the 0-truncated random variable X_T has pmf

Pr(X_T =x) = Pr(X=x)/Pr(X >0), x = 1, 2, . . .

Find the pmf, mean, and variance of the 0-truncated random variable starting from

(a) X ~ Poisson(λ)

(b) X ~ negative binomial (r, p).