Exam Details
| Subject | statistics | |
| Paper | ||
| Exam / Course | m.sc | |
| Department | ||
| Organization | central university | |
| Position | ||
| Exam Date | 2016 | |
| City, State | telangana, hyderabad |
Question Paper
1. A and B are two independent events whose probabilities are strictly greater than zero and strictly less than one. Identify the wrong statement.
A and are independent.
and are independent.
For any event C with 0
2. The number of different arrangements of 10 balls of which m 0 are red and the rest are blue is maximum for m equal to
6.
5.
4.
3.
3. The marks of a class of 100 students are positively skewed, the average marks are so,
more than half of the students in this class have got less than 62% marks.
half of the students in this class have got more than 62% marks.
more than 62 students have got more than 70% marks.
exactly 50 students have got more than 62% marks.
4. The mean and median incomes of 100 households in a locality were both equal to Rs.50, 000, 6 households whose incomes were Rs.28000, 55000, 30000, 52000, 26000, 54000 left the locality and in their places 6 new households joined the locality and their incomes are Rs30000, 40000, 28000, 60000, 62000,72000, with these changes
there is no change in either the mean income or the median income.
the mean and median incomes of the locality have increased but both are still equal.
the median income is unchanged, but the mean income of the locality has increased.
the median income has reduced, but the mean income of the locality has increased.
5. Two coins with probabilities of heads Pi and P2 are tossed simultaneously, and if Pr(both coins show up tails) Pr(both coins show up heads), then Pi +P2 is equal to
1.
3/4.
1/2.
1/4.
6. If X then
15) 0.5.
15) 0.5.
15) 0.5.
15) 0.5.
7. Given in the table is the frequency distribution of a random variable say, X
X 2 4 6 8 Total
Frequency m-1 m-1 2m+3 2m+1 50
the Mode is
not unique.
19.
2 or 4.
6.
8. Which of the following describes a Poisson random variable correctly?
The number of red balls in a collection of 100 balls drawn with replacement from a big bag containing 10,000 red and 20,000 blue balls.
The number of heads that show up when a fair coin(probability of heads is tossed 5 million " times.
The number of people with a super high IQ(Intelligence Quotient), say over 180 in a higWy populous country.
the number of rolls of a fair die till 2 consecutive 6s show up.
9. The total income of all the 100 households in a locality is Rs.25, 000, 000, the expected value of the total income of a simple random sample without replacement of 10 households is
Rs.250, 000
Rs.2500, 000
Rs.25,000
Rs.2500
10. Which of the following is not an observation of a Binomial random variable?
The number of heads that show up in 20 tosses of a biased coin.
The number of times at least one 6 shows up when two dice, one fair and one biased are rolled simultaneously 10 times.
The number of times a red ball is drawn in 5 draws with replacement from a bag containing 50 red and 100 blue balls.
The total number of heads that show up when a fair coin is tossed 10 times and then a biased coin is tossed 6 times.
11. Two fair coins are tossed simultaneously till heads shows up on at least one of the coins, the probability that these coins have to be tossed at least 4 times is
1/64.
1/8.
1/16.
3/256.
12. X the correlation coefficient between X and Y IXI is
-1.
.1.
0.
13. The correlation coefficient between two random variables X and Y is and the variance of Y is four times that of when X is Y is 12, if X is observed to be 12, what can you say about
Can't say anything based on the data given.
Y is equal to 24.
Y will be 24 or 26 with probabilities 1/2 each.
Y is equal to 26.
14. For any independent random variables Xl and X2 whose expected values are zero and variances are 25, identify the incorrect statement:
Xl X2 and Xl X2 are always independent.
V(X1 X2) 50.
V(X1 X2) V(X1 X2).
Cov(X1 X2, Xl X2) 0.
15. P(Z 0.97725, where Z is the standard normal random variable. If the marks in a public exam are normally distributed with mean 45 and standard deviation 10, the proportion of students who got more than 70% marks
is equal to 0.02275.
is more than 0.025.
is less than 0.02275.
can not be determined.
16. In order to estimate the average number of movies seen by individuals in city per month, it was decided to stratify the population into disjoint strata to obtain more reliable estimates, a good criterion to stratify all the people of the city is
Age.
Height.
Weight.
Favourite Colour.
17.
src='./qimages/8045-17.jpg'>
18. Xl and X2 is another random variable whose probability density function is
12x^2(1 0 x 1
0 o.w
identify the correct statement about the probability distributions of Xl and X2.
E(X2).
Xl Pr(0 X2
E(X2).
Xl Pr(0 X2
19. Two heads showed up in n tosses of a fair coin, the maximum likelihood estimate for n is
(a)5or 6.
(b)3or 4.
20. The value of
src='./qimages/8045-20.jpg'>
is equal to 0.
is equal to 2.
is equal to
does not exist.
21. When we say that is a 95% confidence interval for a parameter based on a sample Xl, ... ,X50 of size 50, it means that
the probability that E is 0.95.
the random interval ... ... contains TITA for 95% of all samples of size 50 and 39 and 69 are the observed values of ... and U(XI ... respectively for the given sample.
there is a small probability of .05 that 39 or TITA> 69.
the probability that TITA< 39 is at most 0.05.
22. Which of the following is the most suitable approximation for the number of heads in very large number of tosses of a fair coin:
Poisson.
Negative Binomial.
Normal.
Exponential.
23. Xl, X2,X3 is a random sample from a random variable for which Pr.(X Pi Pr.(X 0 the statistic T(XI Xl +X2 -X3 is
an unbiased and sufficient estimator for p.
an unbiased but not a sufficient estimator for p.
the most efficient estimator for p.
a sufficient but not an unbiased estimator for p.
24. The effectiveness of any treatment for diabetes may be effected by the weight of the patient, so 4 particular treatments were given for 3 months each to 4 patients each randomly in three weight groups -Normal, Heavy, Very Heavy, the percentage decrease in fasting sugar levels was observed in all the patients after 3 months of their respective treatments. This Designed Experiment is a
Completely Randomized Design and 48 patients are required for this study.
Randomized Block Design meant to test whether each treatment is equally effective in each weight group.
RBD for testing whether patients in the different weight groups respond similarly to each of the 4 treatments.
RBD for testing whether each treatment is equally effective on an average across the weight groups.
25. "Ashok secured top grades in at least two of the five courses" is equivalent to
Ashok did not get top grades in three courses.
Ashok did not get top grades in at least three courses.
Ashok secured top grades in all the five courses.
Ashok did not get top grades in at most three courses.
Questions have more than one correct option. For the answer to be right all the correct options have to be marked on the OMR sheet. No credit will be given for partially correct answers. • Questions have only one correct option. • Find the correct answers and mark them on the OMR sheet. Correct answers (marked in OMR sheet) to a question get 3 marks and zero otherwise.
26. The sum of 7 positive real numbers ,a7 is 14, so
their product ala2a3a4a5a6a7 can be 125.
their product ala2a3a4a5a6a7 can not be be 130.
if 5 of a1, ... ,a7 are greater than the sum and the product of the remaining two are both less than 4.
unless all of a1, ... ,a7 are equal, at least one of them is greater than 2.
27. An odd number is twice as likely to be selected as an even number from ... one number is selected, let Band, C be the events that an even number is selected, number selected is a multiple of 3 and that the selected number is a multiple of 5 respectively. Identify the correct statements:
A and B are independent events.
The expected value of the selected number is more than 5.
A and C are independent events.
28. For any pair of iid random variables Xl and X2 that are symmetric about zero, identify the pairs that are identically distributed
X2 and Xl X2·
X2and X2 Xl.
(c)2X1 X2 and Xl 2X2 .
Xl and (X1 •
29. In a family that has three children, it is known that there is at least one girl child, a child is as likely to be a girl as a boy so, the probability that
the other two are boys is more than the probability that the other two are girls.
the other two are boys is equal to the probability that the other two are girls.
there are two girls is equal to the probability that there are boys in this family.
there are more girls is more than the probability that there are more boys in this family.
30. The Lowest,Average and Highest marks obtained by students in two classes A and B are given below:
Class Lowest Average Highest
A 40 55 95
B 35 50 85
If we send the student who got 95, marks from Class to Class and one who got 85 marks from Class to Class we have
reduced the difference in the average marks of the two classes.
increased the variation in marks in at least one of the classes.
reduced the variation in marks in both the classes.
reduced the variance in the marks of all the students.
31. The heights of Indian adult males are normally distributed with mean 167em. and standard deviation 7cm., so
half of the Indian adult males are taller than 167em..
about as many are taller than 184em. as those shorter than 153cm..
if there are 100000 Indian adult males, there may be at most 14 of them who are taller than 188em,.
half of them are shorter than 160cm..
32. Ho and HI are the null and alternate hypotheses regarding a random variable, some observations are made of this random variable, the probability of getting the observations that we have if Ho is true is
0.98 and if HI is true, the probability of getting this data is 0.4,
this implies that Ho is true.
this implies that we should accept Ho because HI is false.
we should accept Ho because data contain significantly more evidence in favour of Ho than for
the p-value for this test will be large.
33. 45 people are employed in a company, the highest income is drawn by 5 of them while 10 each of them draw the second, third, fourth highest and the remaining 10 draw the least income, to estimate the minimum and maximum incomes of this company, it was decided to draw a simple random sample without replacement of 10 employees and use the minimum income and maximum income in the sample as estimators respectively. Identify the correct statements.
the probability of there being an individual with the highest income in the sample is a little less than 1/2.
the sample minimum and maximum incomes are not unbiased estimators of the minimum and maximum wages respectively.
the sample minimum overestimates the minimum income, while the sample maximum underestimates the maximum income.
if the highest, second highest, third highest and fourth highest incomes are and 2 times as much as the minimum income, the expected value of the sample maximum is almost half of the maximum income.
34.
src='./qimages/8045-34.jpg'>
35.
src='./qimages/8045-35.jpg'>
36.The function F defined on R as
0 x 0
1/2 x/2 0 x 1
1 x 1
Then F is
bounded
(b)non-decreasing in x
(c)continuous every where
(d)not continuous at but continuous every where else
37. Consider a 3 x 3 real matrix
2 1 1
A 1 1 1
a 2 2
Rank(A) whatever a maybe.
det(A) not equal to 0 for some choices of a.
Rank(A) whatever a maybe.
there is no a E R. for which Rank(A) 1.
38. There are 5 balls in a bag and it is equally likely that or all 5 of them are red, if two balls are drawn without replacement from this bag, the probability that at least one of them is red is
(a)0.5.
(b)0.6.
(c)O.72.
(d)0.8.
39. All 100 students in a class in UoH know at least one of the three languages, Telugu, Hindi and English. 80 know Telugu, 40 know Hindi and 60 know English, further, 20 know both Telugu and Hindi, 50 know both Telugu and English, while 25 know both Hindi and English, then the number of students who know Telugu and English but not Hindi is
35.
25.
20.
18.
40. Two randomly selected squares are to be removed from a chess board, if every pair to be removed is equally likely, the probability that the two removed squares are neither in the same row, nor in the same column, nor in the same diagonal-main or off is
less than 1/3.
more than but, less than 2/3.
more than but, less than 3/4.
more than 3/4.
41. The probability distribution of a random variable X is Pr(X j regarding its moments,
the first raw moment exists, none of the raw moments of order higher than 1 exist,
the first and second raw moments exist.
all the raw moments exist.
none of the raw moments exist.
42. Select two numbers without replacement from .. assume that every selection of two numbers is equally likely, what is the probability that the product of the selected numbers is at least 20 given that at least one of the selected numbers is a prime number?
15/30.
16/30.
17/30.
18/30.
43. The expected value of a random variable X is 55, and the probability that this random variable takes values between 25 and 85 is 0.98, so
the variance of X could be 20.
the variance of X is 18.
the variance of X is less than 15.
nothing can be said about the variance of X based on the information given.
44. The mean of a Poisson random variable X is then, is
2
3.
less than 1.
more than 3.
45.
src='./qimages/8045-45.jpg'>
46. Xl and X2 are independent exponential random variables, with means 1/2 and 1/3 respectively, then Pr(X1 X2) is
1/5.
2/5.
4/5.
1.
47.
src='./qimages/8045-47.jpg'>
Power 1/2 and Size 1.
Power 1/2 and Size 1/2.
Power 1 and Size 1/2.
Power 1 and Size 0.
48.
src='./qimages/8045-48.jpg'>
49. 2.8,2.1,2.6,2.4,1.9,2.7 is a random sample from the U(TITA,TITA TITA 0 distribution, the maximum likelihood estimate for 0 based on this sample is
unique and is equal to 1.9.
unique and is equal to 2.4.
1.9 or 2.
every number in the interval [1.8,1.9].
50. A is a 5 x 5 real matrix, and its rank is B is another 5 x 5 real matrix in which the first and fifth rows are the fifth and first rows of A respectively, the second and the fourth rows are the fourth and second rows of A respectively, while third row is the third row of so certainly,
Rank(B) Rank(A 3. Rank
Rank(A 5.
Rank(E) Rank(A 4.
Rank(E) Rank(A 5.
A and are independent.
and are independent.
For any event C with 0
2. The number of different arrangements of 10 balls of which m 0 are red and the rest are blue is maximum for m equal to
6.
5.
4.
3.
3. The marks of a class of 100 students are positively skewed, the average marks are so,
more than half of the students in this class have got less than 62% marks.
half of the students in this class have got more than 62% marks.
more than 62 students have got more than 70% marks.
exactly 50 students have got more than 62% marks.
4. The mean and median incomes of 100 households in a locality were both equal to Rs.50, 000, 6 households whose incomes were Rs.28000, 55000, 30000, 52000, 26000, 54000 left the locality and in their places 6 new households joined the locality and their incomes are Rs30000, 40000, 28000, 60000, 62000,72000, with these changes
there is no change in either the mean income or the median income.
the mean and median incomes of the locality have increased but both are still equal.
the median income is unchanged, but the mean income of the locality has increased.
the median income has reduced, but the mean income of the locality has increased.
5. Two coins with probabilities of heads Pi and P2 are tossed simultaneously, and if Pr(both coins show up tails) Pr(both coins show up heads), then Pi +P2 is equal to
1.
3/4.
1/2.
1/4.
6. If X then
15) 0.5.
15) 0.5.
15) 0.5.
15) 0.5.
7. Given in the table is the frequency distribution of a random variable say, X
X 2 4 6 8 Total
Frequency m-1 m-1 2m+3 2m+1 50
the Mode is
not unique.
19.
2 or 4.
6.
8. Which of the following describes a Poisson random variable correctly?
The number of red balls in a collection of 100 balls drawn with replacement from a big bag containing 10,000 red and 20,000 blue balls.
The number of heads that show up when a fair coin(probability of heads is tossed 5 million " times.
The number of people with a super high IQ(Intelligence Quotient), say over 180 in a higWy populous country.
the number of rolls of a fair die till 2 consecutive 6s show up.
9. The total income of all the 100 households in a locality is Rs.25, 000, 000, the expected value of the total income of a simple random sample without replacement of 10 households is
Rs.250, 000
Rs.2500, 000
Rs.25,000
Rs.2500
10. Which of the following is not an observation of a Binomial random variable?
The number of heads that show up in 20 tosses of a biased coin.
The number of times at least one 6 shows up when two dice, one fair and one biased are rolled simultaneously 10 times.
The number of times a red ball is drawn in 5 draws with replacement from a bag containing 50 red and 100 blue balls.
The total number of heads that show up when a fair coin is tossed 10 times and then a biased coin is tossed 6 times.
11. Two fair coins are tossed simultaneously till heads shows up on at least one of the coins, the probability that these coins have to be tossed at least 4 times is
1/64.
1/8.
1/16.
3/256.
12. X the correlation coefficient between X and Y IXI is
-1.
.1.
0.
13. The correlation coefficient between two random variables X and Y is and the variance of Y is four times that of when X is Y is 12, if X is observed to be 12, what can you say about
Can't say anything based on the data given.
Y is equal to 24.
Y will be 24 or 26 with probabilities 1/2 each.
Y is equal to 26.
14. For any independent random variables Xl and X2 whose expected values are zero and variances are 25, identify the incorrect statement:
Xl X2 and Xl X2 are always independent.
V(X1 X2) 50.
V(X1 X2) V(X1 X2).
Cov(X1 X2, Xl X2) 0.
15. P(Z 0.97725, where Z is the standard normal random variable. If the marks in a public exam are normally distributed with mean 45 and standard deviation 10, the proportion of students who got more than 70% marks
is equal to 0.02275.
is more than 0.025.
is less than 0.02275.
can not be determined.
16. In order to estimate the average number of movies seen by individuals in city per month, it was decided to stratify the population into disjoint strata to obtain more reliable estimates, a good criterion to stratify all the people of the city is
Age.
Height.
Weight.
Favourite Colour.
17.
src='./qimages/8045-17.jpg'>
18. Xl and X2 is another random variable whose probability density function is
12x^2(1 0 x 1
0 o.w
identify the correct statement about the probability distributions of Xl and X2.
E(X2).
Xl Pr(0 X2
E(X2).
Xl Pr(0 X2
19. Two heads showed up in n tosses of a fair coin, the maximum likelihood estimate for n is
(a)5or 6.
(b)3or 4.
20. The value of
src='./qimages/8045-20.jpg'>
is equal to 0.
is equal to 2.
is equal to
does not exist.
21. When we say that is a 95% confidence interval for a parameter based on a sample Xl, ... ,X50 of size 50, it means that
the probability that E is 0.95.
the random interval ... ... contains TITA for 95% of all samples of size 50 and 39 and 69 are the observed values of ... and U(XI ... respectively for the given sample.
there is a small probability of .05 that 39 or TITA> 69.
the probability that TITA< 39 is at most 0.05.
22. Which of the following is the most suitable approximation for the number of heads in very large number of tosses of a fair coin:
Poisson.
Negative Binomial.
Normal.
Exponential.
23. Xl, X2,X3 is a random sample from a random variable for which Pr.(X Pi Pr.(X 0 the statistic T(XI Xl +X2 -X3 is
an unbiased and sufficient estimator for p.
an unbiased but not a sufficient estimator for p.
the most efficient estimator for p.
a sufficient but not an unbiased estimator for p.
24. The effectiveness of any treatment for diabetes may be effected by the weight of the patient, so 4 particular treatments were given for 3 months each to 4 patients each randomly in three weight groups -Normal, Heavy, Very Heavy, the percentage decrease in fasting sugar levels was observed in all the patients after 3 months of their respective treatments. This Designed Experiment is a
Completely Randomized Design and 48 patients are required for this study.
Randomized Block Design meant to test whether each treatment is equally effective in each weight group.
RBD for testing whether patients in the different weight groups respond similarly to each of the 4 treatments.
RBD for testing whether each treatment is equally effective on an average across the weight groups.
25. "Ashok secured top grades in at least two of the five courses" is equivalent to
Ashok did not get top grades in three courses.
Ashok did not get top grades in at least three courses.
Ashok secured top grades in all the five courses.
Ashok did not get top grades in at most three courses.
Questions have more than one correct option. For the answer to be right all the correct options have to be marked on the OMR sheet. No credit will be given for partially correct answers. • Questions have only one correct option. • Find the correct answers and mark them on the OMR sheet. Correct answers (marked in OMR sheet) to a question get 3 marks and zero otherwise.
26. The sum of 7 positive real numbers ,a7 is 14, so
their product ala2a3a4a5a6a7 can be 125.
their product ala2a3a4a5a6a7 can not be be 130.
if 5 of a1, ... ,a7 are greater than the sum and the product of the remaining two are both less than 4.
unless all of a1, ... ,a7 are equal, at least one of them is greater than 2.
27. An odd number is twice as likely to be selected as an even number from ... one number is selected, let Band, C be the events that an even number is selected, number selected is a multiple of 3 and that the selected number is a multiple of 5 respectively. Identify the correct statements:
A and B are independent events.
The expected value of the selected number is more than 5.
A and C are independent events.
28. For any pair of iid random variables Xl and X2 that are symmetric about zero, identify the pairs that are identically distributed
X2 and Xl X2·
X2and X2 Xl.
(c)2X1 X2 and Xl 2X2 .
Xl and (X1 •
29. In a family that has three children, it is known that there is at least one girl child, a child is as likely to be a girl as a boy so, the probability that
the other two are boys is more than the probability that the other two are girls.
the other two are boys is equal to the probability that the other two are girls.
there are two girls is equal to the probability that there are boys in this family.
there are more girls is more than the probability that there are more boys in this family.
30. The Lowest,Average and Highest marks obtained by students in two classes A and B are given below:
Class Lowest Average Highest
A 40 55 95
B 35 50 85
If we send the student who got 95, marks from Class to Class and one who got 85 marks from Class to Class we have
reduced the difference in the average marks of the two classes.
increased the variation in marks in at least one of the classes.
reduced the variation in marks in both the classes.
reduced the variance in the marks of all the students.
31. The heights of Indian adult males are normally distributed with mean 167em. and standard deviation 7cm., so
half of the Indian adult males are taller than 167em..
about as many are taller than 184em. as those shorter than 153cm..
if there are 100000 Indian adult males, there may be at most 14 of them who are taller than 188em,.
half of them are shorter than 160cm..
32. Ho and HI are the null and alternate hypotheses regarding a random variable, some observations are made of this random variable, the probability of getting the observations that we have if Ho is true is
0.98 and if HI is true, the probability of getting this data is 0.4,
this implies that Ho is true.
this implies that we should accept Ho because HI is false.
we should accept Ho because data contain significantly more evidence in favour of Ho than for
the p-value for this test will be large.
33. 45 people are employed in a company, the highest income is drawn by 5 of them while 10 each of them draw the second, third, fourth highest and the remaining 10 draw the least income, to estimate the minimum and maximum incomes of this company, it was decided to draw a simple random sample without replacement of 10 employees and use the minimum income and maximum income in the sample as estimators respectively. Identify the correct statements.
the probability of there being an individual with the highest income in the sample is a little less than 1/2.
the sample minimum and maximum incomes are not unbiased estimators of the minimum and maximum wages respectively.
the sample minimum overestimates the minimum income, while the sample maximum underestimates the maximum income.
if the highest, second highest, third highest and fourth highest incomes are and 2 times as much as the minimum income, the expected value of the sample maximum is almost half of the maximum income.
34.
src='./qimages/8045-34.jpg'>
35.
src='./qimages/8045-35.jpg'>
36.The function F defined on R as
0 x 0
1/2 x/2 0 x 1
1 x 1
Then F is
bounded
(b)non-decreasing in x
(c)continuous every where
(d)not continuous at but continuous every where else
37. Consider a 3 x 3 real matrix
2 1 1
A 1 1 1
a 2 2
Rank(A) whatever a maybe.
det(A) not equal to 0 for some choices of a.
Rank(A) whatever a maybe.
there is no a E R. for which Rank(A) 1.
38. There are 5 balls in a bag and it is equally likely that or all 5 of them are red, if two balls are drawn without replacement from this bag, the probability that at least one of them is red is
(a)0.5.
(b)0.6.
(c)O.72.
(d)0.8.
39. All 100 students in a class in UoH know at least one of the three languages, Telugu, Hindi and English. 80 know Telugu, 40 know Hindi and 60 know English, further, 20 know both Telugu and Hindi, 50 know both Telugu and English, while 25 know both Hindi and English, then the number of students who know Telugu and English but not Hindi is
35.
25.
20.
18.
40. Two randomly selected squares are to be removed from a chess board, if every pair to be removed is equally likely, the probability that the two removed squares are neither in the same row, nor in the same column, nor in the same diagonal-main or off is
less than 1/3.
more than but, less than 2/3.
more than but, less than 3/4.
more than 3/4.
41. The probability distribution of a random variable X is Pr(X j regarding its moments,
the first raw moment exists, none of the raw moments of order higher than 1 exist,
the first and second raw moments exist.
all the raw moments exist.
none of the raw moments exist.
42. Select two numbers without replacement from .. assume that every selection of two numbers is equally likely, what is the probability that the product of the selected numbers is at least 20 given that at least one of the selected numbers is a prime number?
15/30.
16/30.
17/30.
18/30.
43. The expected value of a random variable X is 55, and the probability that this random variable takes values between 25 and 85 is 0.98, so
the variance of X could be 20.
the variance of X is 18.
the variance of X is less than 15.
nothing can be said about the variance of X based on the information given.
44. The mean of a Poisson random variable X is then, is
2
3.
less than 1.
more than 3.
45.
src='./qimages/8045-45.jpg'>
46. Xl and X2 are independent exponential random variables, with means 1/2 and 1/3 respectively, then Pr(X1 X2) is
1/5.
2/5.
4/5.
1.
47.
src='./qimages/8045-47.jpg'>
Power 1/2 and Size 1.
Power 1/2 and Size 1/2.
Power 1 and Size 1/2.
Power 1 and Size 0.
48.
src='./qimages/8045-48.jpg'>
49. 2.8,2.1,2.6,2.4,1.9,2.7 is a random sample from the U(TITA,TITA TITA 0 distribution, the maximum likelihood estimate for 0 based on this sample is
unique and is equal to 1.9.
unique and is equal to 2.4.
1.9 or 2.
every number in the interval [1.8,1.9].
50. A is a 5 x 5 real matrix, and its rank is B is another 5 x 5 real matrix in which the first and fifth rows are the fifth and first rows of A respectively, the second and the fourth rows are the fourth and second rows of A respectively, while third row is the third row of so certainly,
Rank(B) Rank(A 3. Rank
Rank(A 5.
Rank(E) Rank(A 4.
Rank(E) Rank(A 5.