PROBABILITY AND STATISTICS
Part-A
1. A card is drawn from a well shuffled pack of playing cards. What is the probability that it is either a spade or an ace?
2. Define Moment Generating Function of a R.V?
3. Find p for a binomial R.V X if n = 6 and if 9P [X=4] = P [X=2].
4. Find the PDF of Y=3X+1 if PDF of X is f(x) = 2x for 0=x=1,0=y=1 & f(x)=0,otherwise.
5. Find the marginal density function of X if the joint PDF of X & Y is
f(x) = 2-x-y for 0 = x =1,0 = y =1 and f(x) = 0, otherwise.
6. State Central Limit Theorem.
7. Define Type I error and Type II error.
8. Write any two applications of ?2 – test.
9. What is the main advantage of Latin square design over randomized block design?
10. Is a 2 x 2 Latin square design possible? Why?
Part –B
* A and B alternatively cut a pack of cards and the pack is shuffled after each cut. If A starts the game and is continued until one cuts a diamond, what are the respective chances of A and B first cutting a diamond. (8)
* Find the MGF of the R.V. X with probability mass function P[X=x] = nCxpxqn-x, x= 0,1,2,…n.(8)
* The contents of urns I,II,III are as follows,
1 white, 2 black, 3 red balls.
2 white, 1 black, 1 red ball.
4 white, 5 black, 3 red balls.
One urn is chosen at random and two balls are drawn from it. They happen to be one white and one red. What is the probability that they come from urns I,II,III? (8)
* A continuous random variable X has the distribution function f(x) given by,
= 0, x=1
= k(x-1)4, 1
= 0, x >3.
Find k, probability density function f(x), P[X<2].(8)
* Find the MGF of the geometric distribution and hence find its mean and variance.(8)
* The mean yield for one acre plot is 662 kilos with standard deviation 32 kilos. Assuming normal distribution, how many one-acre plots in a patch of 1000 plots would you expect to have yield over 700 kilos, below 650 kilos. (8)
* If X is a Poisson variate such that P[X=2] = 9P[X=4] + 90P[X=6], find the mean and variance.(8)
* Subway trains on a certain line run every half hour between mid-night and six in the morning. What is the probability that a man entering the station at a random time during this period will have to wait at least twenty minutes? (8)
* The joint PDF of two R.V. X and Y is P[X=x, Y=y] = (2x+y) / 27 for x = y = 0, 1, 2.
Find the conditional distribution of Y given that X = x.(8)
* The joint PDF of the R.V. X and Y is f(x, y) =
= 2-x-y, 0=x=1, 0=y=1
= 0, elsewhere
Find the correlation co – efficient between X and Y.(8)
* The joint PDF of the R.V. X and Y is f(x, y) =
= 2, 0=x=1, 0=y=x
= 0, elsewhere.
Examine if X and Y are independent or not.(8)
* Let (X, Y) be a two dimensional non-negative continuous R.V. having the joint PDF f(x, y)
= 4xye-(x2+y2) for x=0, y=0
0, otherwise.
Find the density function of U = v X2 + Y2.(8)
* In a referendum submitted to the students to the body at a university, 850 men and 560 women voted. 500 men and 320 women voted favorably. Does this indicate a significant difference of opinion between men and women on this manner at 1% level of significance? (8)
* Two independent samples are chosen from schools A and B a common test is given in a subject. The scores of the students are as follows. (8)
School A: 76 68 70 43 94 68 33
School B: 40 48 92 85 70 76 68 22
* Two random samples drawn from two normal populations are given below:
Sample I: 24 27 26 21 25
Sample II: 27 30 32 36 28 23. Test if the two populations have same variances. (8)
* Do the following data give evidence of the effectiveness of inoculation? (8)
Attacked Not Attacked
Inoculated 20 300
Not Inoculated 80 600.
* The following table shows the lives in hours of four brands of electric lamps.
Brand A 1610 1610 1650 1680 1700 1720 1800
Brand B 1580 1640 1640 1700 1750
Brand C 1460 1550 1600 1620 1640 1660 1740 1820
Brand D 1510 1520 1530 1570 1600 1680
Perform an analysis of variance test the homogeneity of the mean lives of the four brands of lamps. (16)
* Analyze the data given below and interpret the results.
A(13) B(09) C(21) D(07) E(06)
D(09) E(08) A(15) B(07) C(16)
B(11) C(17) D(08) E(10) A(17)
E(08) A(15) B(07) C(10) D(07)
C(11) D(09) E(08) A(11) B(15)
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