Probability and Statistics Model Question Paper (BSc CSIT)

Tribhuvan University
Institute of Science and Technology
Model Question Paper

Bachelor Level/ First Year/ First Semester/ Science                             Full Marks: 60
Computer Science and Information Technology (Stat. 103)           Pass Marks: 24
(Probability and Statistics)                                                                        Time: 3 hours.
Candidates are required to give their answers in their own words as for as practicable.
All notations have the usual meaning.

Group A
Attempt any Two:                                                                                               (2x10=20)

1. Scores of 20 students on a test are summarized below. Compute the followings:  mean, median, mode, Q₁, Q₃, and standard deviation. Comment on the shape of the distribution.
   

   X	95	81	59	68	100	92	75	67	85	79
   Y	71	88	100	91	87	65	93	72	83	91



2. 1. Give the classical definition of probability and comment on the limitations of this definition. Describe how probability is used in testing of hypothesis?
   2. The incidence of a disease in a locality is such that the individuals have a 20% chance of suffering from it. What is the probability out of six individuals chosen at random four or more will suffer from the disease?
3. 1. Write the likelihood function of n independent random sample drawn from a Bernolli distribution f(x, θ). Explain this likelihood function in terms of probability.
   2. Obtain the maximum likelihood estimate of the parameter, proportion of success, in binomial distribution.

Group B
Answer any eight questions:                                                                             (8x5=40)

4. A consulting firm rents cars from three agencies, 20% from agency A, 20% from agency from B, and 60% from agency C. If 10% of the cars from A,  12% of the cars from B, and 4% of the cars from C had bad tires, what is the probability that a car with bad tires rented by the firm came from agency C?


5. The following data shows the number of hours jet aircraft engines have been used and the number of hours required for repair.
   

   Aircraft used in 100 hours (X)	1	2	3	4	5
   Repair time in hours (Y)	10	40	30	80	90

   
   Fit a least square line to estimate mean repair time at X = 4.5.


6. Suppose . Find the probability that X lies (a) between 43 and 54, (b) between 40 and 45 (c) between 60 and 75. If population size is 1000, find the number of units between 60 and 75.


7. A book of 200 pages contains 400 misprints. Selecting pages at random, find the probability (i) 3 misprints in 10 pages, (ii) no misprints in 10 pages, and (iii) more than 2 misprints on a single page.


8. A random sample of 16 observations from normal population has mean of 41.5 and sum of square deviation from sample mean is 135. Find 95% confidence interval for population mean.


9. What are the criteria for good estimators? Show that the sample mean is and unbiased and consistent estimator of a population mean.


10. State and prove the additive law of probability. Hence, derive the additive law when two events are (a) mutually exclusive, and (b) independent.


11. Two discrete random variables X and Y have the following joint distribution:
    
    
    Examine whether X and Y are independent.


12. Give the canonical definitions of t-distribution and its density function. Discuss some of its properties.


13. What do you mean by power of the test? The mean height of 100 individuals is also 66 inches; can it reasonably be regarded as a sample from population with mean height 65 inches and standard deviation 2.6 inches?