MA 2262 PROBABILITY AND QUEUEING THEORY
UNIT I : RANDOM VARIABLES
1. Define (i) random variable (ii) discrete and continuous r.v.s with examples.
2. Let X be a r.v. with E(X) = 1 and E(X(X-1)) = 4. Find Var (X/2) and Var (2 - 3X) .
3. The p.d.f. of the samples of the amplitude of speech wave forms is found to decay exponentially at rate, so the following p.d.f. is proposed:. Find the constant ‘c’, P(, E(X) and MGF.
4. If a r.v. X has its cdf given by , find (i) ‘c’ (ii) p.d.f. of X (iii) P(1 < X < 2)
5. If the density function of a continuous r.v. X is . Find ‘a’ and the c.d.f. of X.
6. If the moments of a r.v. X are given by E(X r) = 0.6; r = 1,2,3,…show that P(X=0) = 0.4, P(X=1) = 0.6, P(X≥2) = 0.
7. A continuous r.v. X has its p.d.f. given by. Find the value of ‘c’ and the distribution function.
8. The probability of an infinite discrete distribution is given by P(X=j) = , j = 1,2,…. Find the MGF, mean and variance of the distribution. Also find P( X is even ), P( X ≥ 5 ) and P( X is divisible by 3 ).
9. If the probability that a communication system will have high fidelity is 0.81 and the probability that it will have high fidelity and high selectivity is 0.18, what is the probability that a system with high fidelity will also have high selectivity?
10. Find the MGF, mean and variance for the following distributions:
(i) Binomial (ii) Poisson (iii) Geometric (iv) Negative Binomial (v) Uniform (vi) Exponential (vii) Gamma Distributions.
11. Find the mean and variance of Weibull Distribution.
12. A discrete r.v. X has the following probability distribution:
Values of X : x / 0 / 1 / 2 / 3 / 4 / 5 / 6 / 7P(x) / 0 / a / 2a / 2a / 3a / a2 / 2a2 / 7a2 + a
Find (i) the value of ‘a’ (ii) P( X < 6 ), P( X ≥ 6 ), P( 0 < X < 4 ), P( ) and P( X>4 / X ≥ 2) (iii) cdf of X.
13. If a r.v. X has the p.d.f. , obtain (i) P( X 1 ) (ii) P() and (iii) P( 2X+3 > 5 )
14. Check whether the following are pdf s : (i) (ii)
15. A continuous r.v. X has p.d.f. f(x) = 3x2, 0 < x < 1. Find k and α such that (i) P( X < k) = P( X > k ) and
(ii) P( X > α ) = 0.1
16. A r.v. X has pdf . Obtain the MGF, mean, variance, P( X 2 ). Also find the first four moments about the origin.
17. Find the MGF, mean and variance of a r.v. X having p.d.f. .
18. A discrete r.v. X has MGF MX(t) = . Find E(X), Var(X) and P( X = 2 ).
19. State and prove any 2 properties of MGF.
20. State and prove the additive property of (i) Poisson distribution (ii) exponential distribution with same parameter λ.
21. If X is a binomially distributed r.v. with E(X) = 2 and Var(X) = 4/3, find P(X = 5).
22. Prove that the Poisson distribution is a limiting form of the Binomial Distribution when n → , p → 0 and np remains constant.
23. If X, Y are independent r.v.s each having a geometric distribution, show that the conditional distribution of X given X+Y is uniform.
24. State and prove the memoryless property of (i) Exponential distribution (ii) Geometric distribution.
25. Find the recurrence relation for the moments of the Poisson distribution and hence find the mean, variance and third central moment.
26. If the probability of success is 0.09, how many trials are required to have a probability of atleast one success as 1/3 or more?
27. A communication system consists of n components, each of which will independently function with probability p. The total system will be able to operate effectively if atleast one half of its components function. For what values of p is a 5-component system more likely to operate effectively than a 3-component system?
28. Past experience shows that 2% defective is the process capability of producing fuses in a factory. Obtain the probability that there will be more than 5 defective fuses in a box of 200 fuses.
29. X follows a Poisson distribution such that P( X = 2 ) = P( X = 1 ). Find P( X = 0 ).
30. The no. of monthly breakdowns of a computer is a r.v. having a Poisson distribution with mean equal to 1.8. Find the probability that this computer will function for a month (i) without a breakdown (ii) with only one breakdown.
31. Suppose that the no. of telephone calls coming into a telephone exchange between 10 am and 11 am, say X1, is a Poisson variate with parameter 2. Similarly, suppose the no. of calls arriving between 11 am and 12 noon, say X2, is a Poisson variate with parameter 6. If X1 and X2 are independent, what is the probability that more than 5 calls come between 10 am and 12 noon?
32. Six coins are tossed 6400 times. What is the probability of getting 6 heads 10 times?
33. The probability of a student passing a subject is 0.8. What is the probability that he will pass the subject (i) on his third attempt (ii) before his third attempt?
34. Suppose that a trainee soldier shoots a target in an independent fashion. If the probability that the target is shot on any one shot is 0.7. What is the probability that (i) the target would be hit on the tenth attempt? (ii) it takes him less than 4 shots to hit the target? (iii) it takes him an even no. of shots to hit the target?
35. A pediatrician wishes to recruit 5 couples each of whom is expecting their first child to participate in new natural child birth regimen. If the probability that a randomly selected couple agrees to participate is 0.2, what is the probability that atmost 15 couples must be asked before 5 are found who agree to participate?
36. A die is cast until 6 appears. What is the probability that it must be cast more than 5 times?
37. If the probability is 0.10 that a certain kind of measuring device will show excessive drift, what is the probability that the fifth measuring device tested will be the first to show excessive drift? Find its expected value also.
38. Find the probability that in tossing 4 coins one will get either all heads or all tails for the third time on the seventh toss?
39. In a company 5% defective components are produced. What is the probability that atleast 5 components are to be examined in order to get 3 defectives?
40. Trains arrive at a station at 15-minute intervals starting at 4 am. If a passenger arrives at the station at a time that is uniformly distributed between 9.00 and 9.30 am, find the probability that he has to wait for the train for (i) less than 6 minutes (ii) more than ten minutes.
41. If X is uniformly distributed in [-2,2], find (i) P(X<0) (ii) P( 1/2 )
42. The time required to repair a machine is exponentially distributed with parameter ½. What is the probability that a repair takes atleast 10 hours given that its duration exceeds 9 hours?
43. Suppose that the number of miles that a car can run before its battery wears out is exponentially distributed with an average value of 10,000 miles. If a person desires to take a 5000-mile trip, what is the probability that he/she will be able to complete the trip without having to replace the car battery?
44. Suppose that the lifetime of a certain kind of an emergency back-up battery (in hours) is a r.v. X having a Weibull distribution with parameters α = 0.1 and β = 0.5, find (i) the mean lifetime of these batteries (ii) the probability that such a battery will last for more than 300 hours.
45. On an average 1.3 gamma particles/millisecond come out of a radioactive substance, determine (i) Var(X)
(ii) P(X>0)
46. The daily consumption of bread in a hostel, in excess of 2000 loaves, is approximately Gamma distributed with parameters α = 2 and β = 1000. The hostel has a daily stock of 3000 loaves. What is the probability that the stock is insufficient on a day?
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