# Andhra University BE/B.Tech Probability Statistics and Queuing Theory Previous Paper 2010

First Question is compulsory

Answer any FOUR from the remaining questions.

All questions carry equal marks.

Answer ALL parts of any question at one place.

- (a) Define conditional probability.

(b) What is geometric probability?

(c) What is probability density function?

(d) Define moment generating function.

(e) Define parameter with an example.

(f) What is meant by standard error?

(g) What is calling population in a queue.

2. (a) A candidate is selected for interview for three posts. For the first post there are 6 candidates, for the second post there are 9 candidates and for the third there are 5 candidates. What are the chances for his getting at least one post?

(b) The probability that a student Mr. X passed mathematics is 2/3, the probability that he passes statistics is 4/9, if the probability of passing at least one subject is 4/5, what is the probability that Mr. X will pass both the subjects?

3. (a) State and prove the Bayes theorem.

(b) Three balls are drawn without replacement from an urn containing 4 red and 6 white balls. If X is a random variable which denotes the total number of red balls drawn, construct a table showing the probability distribution of X. Also find the expectation.

4. (a) Consider the probability density function

f(x)= 0.5(x+1), -1≤ x ≤1

= 0, elsewhere

Calculate (i) Mean and (ii) Median.

(b) Given the joint density

f ( x, y) =x+y,0 <x<1, 0<y<1

=0, elsewhere

Find the conditional; density of Y given X=x.

5. (a) Obtain the moment generating function of Poisson distribution and hence fins its mean and variance.

(b) Small stones are collected and weights are assumed to be normal. It is found that 5% of the stones are under 30 gm and 80%are under 50 gm. What are the mean and standard deviation of the distribution?

6. (a) Obtain a second degree regressed polynomial from the following data :

X: 0 1 2 3 4

Y: 1 1.5 2.6 4.2 6.8

(b) The marks obtained by 15 students in an examination have a mean 60 and variance 30. Find 99% confidence internal for the mean of the population marks, assuming it to be normal.

7. (a) A die was thrown 500 times and six resulted 100 times. Do the data justify the hypothesis of an unbiased die?

(b) At a one-man barber shop, customers arrive according to the poisson distribution with a mean arrival rate of 4 per hour and his hair cutting time was exponentially distributed with an average hair cut taking 12 minutes. There is no restriction in queue length. Calculate the following .

(i) Fluctuations of the queue length

(ii) Probability that there is at least 5 customers in the system.

8. Write short notes on the following:

(a) Sampling distribution

(b) Test for independence

(c) Properties of exponential distribution.

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