AAOC GC111 Probability & Statistics

8/18/2019 AAOC GC111 Probability & Statistics

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Comprehensive Examination

BITS -PILANI, GOA CAMPUSSecond Semester, 4 -2005

AAOC GCl11 Probability and Statistics

Date: 10.05.05 Monday) TIME: Three Hours MM:120

Instructions: a) There are two parts in this question paper, Part A and Part B. Answer them in

separate answer books provided. b) At the back of the cover page of the answer book make an index of the questions

attempted and the page numbers. c) Start a new question from a new page

Useful Data

t s with 24 degrees of freedom = 2.797t swith 25 degrees of freedom = 2.787

PART-A

1 a). A person has three coins A, B and C: A is unbiased; the probability that a head willhappen when B is tossed is 2/3; the probability that a head will happen when C is tossedis 1/3. If one of the coins, chosen at random is tossed three times, giving a total of twoheads and one tail, fmd the probability that the chosen coin was A.

b). The probabilities of n independent events are PI, 1 2, ... Pn. Find an expression for theprobability tha~ast one of the events will happen.

c). In the year 2006 there will be three candidates for the post of Director; Dr. John, Dr.Philips and Dr. Peter, whose chances of getting the appointment are in the proportion4:2:3 respectively. The probability that Dr. John will abolish coeducation in the Instituteis 0.3. The probability that Dr. Philips and Dr. peter doing the same thing are 0.5 and 0.8respectively. What is the probability that coeducation will be abolished in the Institute?

[7+3+5=15]2 If the probability of a random variable is given by:

{

:Xl/2

I x = kxe-, 0 < x < co0, elsewhere.

a). b). c).

Find the value of k.Find the expectation of X.Find the distribution function of X. [4+7+4 = 15]

3 a) Given a random variable having normal distribution with Jl = 16.2and ci = 1.5625.Findthe probabilities that it will take on a value: i) Greater than 16.8; ii) Less than 14.9; Hi) Between 13.6 and 18.8.

z -0.48 1.04 2.08 2.33 2.325 2.5 2.575

F z) 0.3156 0.8508 0.9812 0.990] 0.99 0.9938 0.995

r 0.849 0.0696 0.871.25 0.069 1.333


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-.(b) If a random sample of size n is taken from a population having mean J. and variance j2.

then show that:

(i) X is a random variable whose mean is J. .(ii) For samples from infinite population, the variance of this distribution is (j2/n.

[6+4+5=15J

4. Suppose that the durability of a paint (in years) is a random variable, say X, has anexponential distribution with mean r3 = 2 i.e. the density for X is given byf(x) = (l/2)e-x/2,x> 0; f(x) = 0, else where.Suppose that 5 houses are painted at the same time, given random numbers .73, .25, .84,.33, .87, simulate the durability of the paints of the five houses and calculate thefollowing.(a) The time of first failure of paints.(b) The time of last failure of paints.(c) The simulated average durability of5 paints. [5+5+5=15]

P RT B

(b)

The mean of a random sample of size n = 25 is used to estimate the mean of an infinite ·population with standard deviation (j = 2.4.Whatcan we assert about the probabilitythatthe error will be less than 1.2, if we use:(a) Chebyshev's theorem.(b) The Central limit theorem. [7+8=15J

(a) A random sample of size 25 from a normal population has mean X = 47.5 andastandard deviation s = 8.4. Does this information tend to support or refute the claim that

the mean of the population is J. = 42.1?(b) If independent random samples of size nl = n2 = 8 come from a normalpo~Ett on of the same variance, what is the probability that either sample variance willbeJiiii1estslarge as the other. [7+8=15JAccording to norms established for a mechanical aptitude test, persons who are 20 yearsold should average 73.2 with standard deviation of 8.6. If 45 randomly selected personsof that age average 76.7. Test the null hypothesis J. = 73.2 against the alternatehypothesis J. > 73.2 at the 0.01 level of significance.It is desired to test the null hypothesis J. = 100 pounds against the alternate hypothesisJ. < 100 pounds on the basis of a random sample of size n = 40 from a population with ~(j = 12.For whatvalueof X mustthe nullhypothesisbe rejectedif the probabilityoftype I error is to be a = 0.01? [8+7=15JIn an Institute the data regarding the weights (X) and heights (Y) of 22 students is asunder:

I> =102.4, Lx2 =483.3, LY = 422.0, Ly2 = 8896.0 andLX) = 1921.9.

(a) Fit a straight line to the given data by the method of least squares and use it topredict the height one can expect when the weight is x = 3.8.(b) Calculate r, the correlation coefficient.(c) Find 99 confidence limits for p. [8+3+4=15]

5.

6.

7(a)

8.


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4;:;;Jj~~ Ii{j?:1;lfJPI~ _ _1:Y 0 and X2 > O~f(XI X2) =0, otherwise,

where kl and k2 are negative constants such that V(Xd = 1.(a) Determine the marginal densities fl (Xl) and 2(X2) in terms of kl and k2.(b) Determine the constants kl and k2.

z 1.645 1.695 1.96 2.4 2.325 2.575 2.666F(z) 0.95 0.955 0.975 0.9918 0.99 0.995 0.996

r 1.666591.97501 IZ 0.935 0.967 I


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A4. (a) Suppose that Xl and X2 are independent random variables with same mean 20and variance 5. Find E 3X1 + X2 + 5)2).

(b) Suppose that a continuous random variable X has a uniform distribution withmean 7 and variance 48. Find the constants a > 0 and b such that Y = aX +b hasmean 10 and variance 3.

P RT B

BI. (a) The chi-square distribution with 10 degrees of freedom is given by

{

lxe-x/2 x > 0f x =]0 ,

0, x ~ O.

Find the probability that the variance of a random sample of size 11 from anorma.l population with a = 20 exceeds 200.

(b) Let X be a random variable having probability density function

{i 0

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AAOC GC111 Probability & Statistics