PROBABILITY AND STATISTICAL APPLICATIONS

JAN 2010 MCA I-SEM QUESTION PAPERS(JNTUK-REGULAR)

Time: 3 Hours Max Marks: 60

Answer any FIVE questions All questions carry EQUAL marks

1. (a) State and Prove the Bayes Theorem.

(b) A and B enter into a bet according to which ‘A’ will get Rs.200 if it rains on that day

and will lose Rs.100 if it does not rain. The probability of raining on that day is 0.7.

What is mathematical expectation of A?

2. (a) Define Binomial distribution and derive the first four moments of a binomial

distribution

(b) The diameter of an electric cable is assume to be a continuous variate with P.d.f.

f(x) = 6x(1-x),

0 x 1. Verify that the above is P.d.f. also find the mean and variance

3. Define a normal Random Variable & derive the properties of normal distribution

4. (a) Derive the Characteristic function of a Poisson distribution

(b) State & prove the Central limit theorem

5. (a) Define a Student ‘t’. Explain the t-test for single mean .

(b) A sample of 400 male students is found to have a mean height of 171.38 cms . Can it

Reasonably regarded as a sample from a large population with mean height 171.17 cms

and standard deviation 3.30 cms?(use 5% level of significance)

6. (a)Explain the F-test for equality of population variances.

(b)Elapsed times for a synthetic job were measured on two different computer systems.

The sample sizes for the two cases were 15 each and the sample means and sample

Variances were computed to be

x =104 seconds y=114 seconds

Sx2 =290 Sy2=510

Test the hypothesis that the population means μx=μy, against the alternative μx < μy.

7. (a) Explain the principle of Least-Squares.

(b) Fit a Second degree curve for the following data

X : 2 3 4 5 6 7

Y: 5 9 18 26 35 50

8. (a)Explain the role of P-charts in statistical quality control

(b) Explain the characteristics of M/M/1 model

JAN 2010 MCA I-SEM QUESTION PAPERS(JNTUK-SUPPLY)

Time: 3 Hours Max Marks: 60

Answer any FIVE questions All questions carry EQUAL marks



1. (a) If the probability that a communication system will have high fidelity is 0.91 and the

probability that it will have high fidelity and selectivity is 0.17. What is the probability

that a system with high fidelity will also have high selectivity?

(b) State and prove Bayes theorem.



2. (a) A continuous random variable has the probability density function

Determine (i) k (ii) Mean (iii) Variance

(b) Find the mean and variance of the uniform probability distribution

given by f(x)=1/n for x=1,2,3,…..n



3. (a) Find the Poisson approximation to the binomial distribution.

(b) A random sample of size 100 is taken from an infinite population having mean 76

and variance 256. What is the probability that sample mean lies between 75 and 78.



4. (a) A normal population has a mean of 0.1 and standard deviation of 2.1. Find the

probability that mean of a sample of size 800 will be negative?

(b) A random sample of size 36 from a normal population has the mean 47.5 and standard

deviation 8.4. Doe this information support or refuse the claim that mean of the

population is 42.1.



5. (a) Describe the method of maximum likelihood for the estimation of unknown

parameters. State the important properties of maximum likelihood estimators.

(b) A coin is tossed 950 times and head turned up 180 times. Is the coin biased?



6. What is meant by (a) a test of null hypothesis? (b) Type I and type II errors (c) Explain

the terms one-tail and two-tail tests?



7. (a) In a random sample of 400 industrial accidents, it was found that 231 were due to

least unsafe working conditions. Construct a 99% confidence interval for the

corresponding proportion.

(b) Obtain a relation of the form y= a.b x for the following data by the method of least

squares.

x  2     3     4       5       6

y 8.4 15.1 33.1 65.2 127.4



8. (a) The following data pertain to the number of jobs per day and the central processing

unit time required.

No. of jobs  1  2  3  4  5

CPU time     2  5  4  9 10

Fit a straight line. Estimate the mean CPU time at x= 3.5

(b) Find the correlation coefficient of the following data

x 10 12 18 24 23 27

y 12 20 12 25 35 10





JAN 2010 MCA I-SEM QUESTION PAPERS(JNTUK-SUPPLY)

Time: 3 Hours Max Marks: 60

Answer any FIVE questions All questions carry EQUAL marks



1. a) Two aeroplanes bomb a target in succession. The probability of each correctly scoring

a hit is 0.3 and 0.2 respectively. The second will bomb only if the first misses the target.

Find the Probability that (i) target is hit (ii) both fails to score hits?

b) State and prove Baye’s theorem?



2. a) Define random variable, discrete probability distribution, continuous probability

distribution and Cumulative distribution?

b) A random variable X has the following probability function:

X 4 5 6 8

P(x) 0.1 0.3 0.4 0.2

Determine (i) Expectation (ii) Variance (iii) Standard Deviation?



3. a) Fit a Poisson distribution for the following data and calculate the expected frequencies?

x 0 1 2 3 4

f(x) 109 65 22 3 1

b)If the masses of 300 students are normally distributed with mean 68kgs and standard

deviation 3kgs, how many students have masses

(i) Greater than 72kg

(ii) Less than or equal to 64kg

(iii) Between 65 and 71kg inclusive?



4. a) A random sample of size 100 is taken from an infinite population having the mean =

76 and the variance 2 = 256. What is the probability that will be between 75 and 78?

b) The mean voltage of battery is 15 and S.D. is 0.2. Find the probability that four such

batteries connected in series will have a combined voltage of 60.8 or more volts?



5. a) Experience had shown that 20% of a manufactured product is of the top quality. In one

day’s production of 400 articles only 50 are of top quality. Test the hypothesis at 0.05

level?

b) An ambulance service claims that it takes on the average less than 10 minutes to reach

its destination in emergency calls. A sample of 36 calls has a mean of 11 minutes and the

variance of 16 minutes. Test the significance at 0.05 level?



6. a) In one sample of 8 observations the sum of the squares of deviations of the sample

values from the sample mean was 84.4 and in the other sample of 10 observations it was

102.6. Test whether this difference is significant at 5% level?

b) Find the maximum difference that we can expect with probability 0.95 between the

means of samples of sizes 10 and 12 from a normal population if their standard

deviations are found to be 2 and 3 respectively?



7. a) Obtain the rank correlation coefficient for the following data

X 68 64 75 50 64 80 75 40 55 64

Y 62 58 68 45 81 60 68 48 50 70

b) Consider the following data on the number of hours which 10 persons studied for a test

and their scores on the test:

Hours Studied(x)   4   9 10 14   4  7  12 22  1 17

Test Score (y)     31 58 65 73 37 44 60 91 21 84



8. What is meant by Statistical Quality Control? The following data provides the values of

sample mean and the Range R for ten samples of size 5 each. Calculate the values for

central line and control limits for mean-chart and range-chart, and determine whether the

process is in control.

Sample

No              1     2      3      4      5     6     7       8     9     10

Mean        11.2 11.8 10.8 11.6 11.0 9.6 10.4  9.6 10.6 10.0

Range (R)    7     4      8      5      7    4      8       4    7      9