ACADEMIC YEAR: 2012 – 2013 (EVEN SEMESTER)
SUBJECT: MA2263 PROBABILITY AND STATISTICS
WORKSHEET / SEMESTER: IV (COMMOM TO CHBT)
UNIT I: RANDOM VARIABLES
1.State any two properties of the cumulative density function of a random variable X.
2.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.
3.Let X and Y be random variables with and Find
4.Let X and Y be independent random variables with and Find.
5.Check whether the following are pdf s: (i) (ii) .
6.If a r.v. X has the p.d.f. , obtain (i) P( X < 1 ) (ii) P() and (iii) P( 2X+3 > 5)
7.A continuous random variable has the probability density function. Find the value of k and also find .
8.The distribution function of a r.v X is given by Find the density function, mean and variance of X.
9.If a r.v. X has its cdf given by , find (i) ‘c’ (ii) p.d.f. of X (iii) P(1 < X < 2)
10.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(Xα) = 0.1
11.The diameter of an electric cable X is continuous random variable with p.d.f Find the (i) value of K (ii) the c.d.f of X (iii0 and (iv) a number b such that
12.A continuous r.v. X has its p.d.f. given by. Find the value of ‘c’ and the distribution function.
13.A continuous r.v X has a pdf Find k, mean and variance.
14.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.
15.Find the first four moments about the origin for a random variable X having the p.d.f
16.The probability of an infinite discrete distribution is given by, x = 1,2,…. Find the MGF, mean and variance of the distribution. Also find P( X is even ), P( X ≥ 4 ) and P( X is divisible by 3).
17.A r.v. X has pdf . Obtain the MGF, mean, variance, P(X > 2). Also find the first four moments about the origin.
18.A continuous r.v X has the pdf f(x) given by Find the value of C and mgf of X.
19.Prove that the moment generating function of the sum of a number of independent r.vs is equal to the product of their respective moment generating functions.
20.Find the MGF, mean and variance of a r.v. X having p.d.f. .
21.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.
22.Ten coins are thrown simultaneously. Find the probability of getting at least seven heads.
23.A discrete r.v. X has MGF MX(t) =. Find E(X), Var(X) and P(X = 2).
24.Find p for a binomial r.v X if n = 6 and if
25.If X is a binomially distributed r.v. with E(X) = 2 and Var(X) = 4/3, find P(X = 5).
26.The random variable X, the number of radar signals properly identified in a 30-minute period, is a binomial random variable with parameters. Find the probability that at most seven signals will be identified correctly, MGF, mean and variance.
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.Cotton linters used in the production of rocket propellant are subjected to a nitration process that enables the cotton fibers to go into solution. The process is 90% effective in that the material produced can be shaped as desired in a later processing stage with probability 0.9. What is the probability that exactly 20 lots will be produced in order to obtain the third defective lot?
29.Six coins are tossed simultaneously. What is the probability of getting (i) 2 heads (ii) atleast 2 heads and (iii) more than 2 heads?
30.Six coins are tossed 6400 times. What is the probability of getting 6 heads 10 times?
31.X follows a Poisson distribution such that P(X = 2) = P(X = 1), determine P(X = 0).
32.If X is a Poisson variate such that find the mean and variance.
33.Suppose the number of accidents occurring weekly on a particular stretch of a highway follows a Poisson distribution with mean 3. Calculate the probability that there is atleast on accident this week.
34.Prove the reproductive property of independent Poisson random variables. Hence find the probability of ‘5’ or more telephone calls arriving in a ‘9’ minute period in a college switch board, if the telephone calls that are received at the rate of ‘2’ every 3 minutes follow a Poisson distribution.
35.The average number of traffic accidents on a certain section of a highway is two per week. Assume that the number of accidents follow a Poisson distribution. Find the probability of no accidents in a 2 week period.
36.Prove that the Poisson distribution is a limiting form of the Binomial Distribution when n → , p → 0 and np remains constant.
37.Find the recurrence relation for the moments of the Poisson distribution and hence find the mean, variance and third central moment.
38.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.
39.The white blood-cell count of a healthy individual can average as low as 6000 per cubic millimeter of blood. To detect a white-cell deficiency, a 0.001 cubic millimeter drop of blood is taken and the number of white cells X is found. How many white cells are expected in a healthy individual? If at most two are found, is there evidence of a white-cell deficiency?
40.In manufacturing electronic circuits, ceramic plates are drilled to provide pathways called “vias” from one surface to another. A typical plate is about the size of a playing card and may require 10,000 vias each as a pinpoint. In the past these vias were drilled using diamond drills. New technology uses lasers to produce these precisely positioned pathways. Suppose that the probability of incorrectly positioning a via is only. What is the probability that a randomly selected plate will have no improperly positioned vias?
41.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?
42.State and prove the memoryless property ofGeometric distribution.
43.If the probability that an applicant for a driver licence will pass the road test on any given trial is 0.8. What is the probability that he will finally pass the test in fewer that 4 trials?
44.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.
45.If X is uniformly distributed in [-2,2], find (i) P(X<0) (ii) P( 1/2 ).
46.If X is uniformly distributed over the interval compute
47.If the m.g.f of a continuous random variable is (e5t-e4t)/t, t0, what are the mean and variance?
48.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.
49.Suppose that during a rainy season in a tropical island the length of the shower has an exponential distribution, with average 2 minutes. Find the probability that the shower will be there for more than three minutes.
50.State and prove the additive property of exponential distribution with parameter λ.
51.State and prove the memoryless property of Exponential distribution.
52.The time (in hours) required to repair a machine is exponentially distributed with parameter (i) what is the probability that the repair time exceeds 2h? (ii) What is the probability that a repair takes atleast 11 hours given that its duration exceeds 8 hours?
53.The daily consumption of milk in excess of 20000 gallons in a town is approximately exponentially distributed with mean 3000. The town has a daily stock of 35000. What is the probability that of two days selected at random, the stock is insufficient on both days?
54.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?
55.In a certain city, the daily consumption of electric power in millions of kilowatt-hours can be treated as a random variable following Gamma distribution with parameters and If the power plant of this city has a daily capacity of 12 million kilowatt-hours, what is the probability that this power supply will be adequate on any given day?
56.The daily consumption of bread in a hostel in excess of 2000 loaves is approximately Gamma distributed with parameters =1/1000 and =2. The hostel has a daily stock of 3000 loaves. What is the probability that the stock is insufficient on any given day?
57.The daily consumption of milk in a city in excess of 20,000 liters is approximately distributed as an Gamma variate with the parameters The city has a daily stock of 30,000 liters. What is the probability the stock is insufficient on a particular day?
58.Find the mean and variance of Weibull Distribution.
59.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.
60.If the life X (in years) of a certain type of car has a Weibull distribution with the parameter find the value of the parameter , given that probability of the life of the car exceeds 5 years is . For these values of obtain the mean and variance of X.
H