Institute of Actuaries of India 2009 CT-3 Probability and Mathematical Statistics ( and) - Question Paper
INSTITUTE OF ACTUARIES OF INDIA
EXAMINATIONS 19th May 2009 Subject CT3 - Probability & Mathematical Statistics Time allowed: Three Hours (10.00 - 13.00 Hrs)
Total Marks: 100
INSTRUCTIONS TO THE CANDIDATES
1. Please read the instructions on the front page of answer booklet and instructions to examinees sent along with hall ticket carefully and follow without exception
2. Mark allocations are shown in brackets.
3. Attempt all questions, beginning your answer to each question on a separate sheet. However, answers to objective type questions could be written on the same sheet.
4. In addition to this paper you will be provided with graph paper, if required.
AT THE END OF THE EXAMINATION Please return your answer book and this question paper to the supervisor separately.
Q 1) A civil engineer monitors water quality by measuring the amount of suspended solids in a sample of river water. Over 11 days, she observed
14 12 21 28 30 63 29 63 55 19 20 suspended solids (parts per million)
a) Draw a dot diagram. (1)
b) Find the median and the mean. Locate both on the dot diagram. (3)
c) Find the sample standard deviation. (2)
[6]
Q 2) The mean weight of 150 students in a class is 60 kgs. The mean weight of boys in the class is 70 kgs and that of girls is 55 kgs.
Find the number of boys and girls. [3]
Q 3) A pair of events A and B cannot be simultaneously mutually exclusive and independent.
Prove that if P(A) > 0 and P(B)> 0 , then
a) If A and B are mutually exclusive, they cannot be independent.
b) If A and B are independent, they cannot be mutually exclusive. [2]
Q 4) Generate three random samples from Binomial B(3, 0.4) using the following values from U(0,1)
0.196 0.351 0.975 [3]
Q 5) a) State the law of total probability for exclusive events and Bayes theorem. (2)
b) A shop keeper buys a particular kind of light bulbs from three manufacturers. A,B and C she buys 30% of her stock from A, 45% from B and 25% from C. In the past, she has found that 2% of Cs bulbs are faulty whereas only 1% of A s and Bs are. Suppose that she chooses a bulb at random and it is faulty, what is the probability that it was one of Cs bulbs. (3)
Q 6) A random variable X has pdf
X2
I f(x) =jX e2c2; a > 0,0 < x <
a
a) Find the Inter Quartile Range (IQR). (2)
b) Show that the ratio of IQR to standard deviation of X is free from a. (4)
Suppose that for a given population with a= 8.4 square inches. One wants to test the null hypothesis H0 : /u= 80.0 square inches against the alternative H] : 80 square inches on the basis of a random sample of size n = 100.
If the null hypothesis is rejected for X < 78.0square inches, what is the probability of type I error? Find the power at /u= 82.
Define compound distribution. (1)
Q 8)
The number of claims N, which arises in a year from a group of policies, has negative binomial distribution
(n + 2 3 P(N = n) = (0.9)3(0.1) nn = 0,1,2,...
I n J
The claim amounts (in Rs.1000s) are independent and identically distributed as gamma (6,2) and also independent of N. Let Y be the total claim amount arising from these policies.
a) Obtain an expression for the mgf of Y (You may assume the mgf of negative binomial and gamma). (3)
b) Compute the standard deviation of Y. (2)
[6]
Let X and Y have joint pmf
Q 9)
p(x,y) =----- if y = 0,1,...,x; x = 0,1,...; V> 0; 0 < p < 1 y!( x _ y)!
a) Find marginal pmfs of X and Y. (4)
b) Find conditional distribution of Y for a given X, conditional distribution of X for a given Y and comment on these results. (4)
c) Find E[Y/X=x] and E[X/Y=y] (2)
[10]
Q 10)
Let X], X2,..., Xn be a random sample from U(-9,9), 9> 0.
a) Find the maximum likelihood estimator of 9. (4)
b) Find the moment estimator of 9 (2)
c) State CRLB? Explain why CRLB cannot be applied in this case. (2)
d) Using the data below compute ML estimate moments estimate of 9
4.0 4.5 -3.8 -1.4 4.3 2.8 1.2 2.7 3.1 -2.2 (2)
[10]
Q 11) The yields of tomato plants grown using different types of fertilizers are given below.
Fertilizer Yield (in kgs)
|
X |
3.5 |
4.0 |
3.8 |
4.1 |
4.4 |
|
Y |
4.7 |
5.0 |
4.5 |
5.3 |
4.6 |
|
Z |
3.6 |
3.9 |
4.2 |
4.1 |
4.0 |
a) Is there evidence that the fertilizers produce different yields? Test at 5% level. State
the hypothesis and assumptions you make. (4)
b) Irrespective of the conclusions in (a) above, if X and Z are chemical fertilizers and Y is a natural fertilizer, examine whether there is evidence to suggest that the average effect of the two chemical fertilizers are inferior to natural fertilizers regarding the yield. (3)
Q 12) State the postulates of Poisson process. If the number of accidents in a town follows a Poisson process with a mean of 2 per day and the number X, of people involved in the accident has the distribution
p( xi=k)=k=i-2-''
Obtain the mean and variance of the number of people involved in accidents per week. [5]
Q 13) An experiment was conducted on 20 identical square metal sheets, in order to study the increases in (a) their length (X in cms), (b) their breadth (Y in cms) and (c) the difference between their length and breadth ((X-Y) in cms), when they were subjected to a heat process.
The following summary was observed.
Sample variance of X (s\)= 1.2011 ; Sample variance of Y (sj)= 2.2958 Sample variance of (X-Y) (sX_Y})= 1.8673
a) Calculate the sample correlation coefficient between X and Y. (1)
b) Construct the 95% CI for the population correlation coefficient p, on assuming bivariate normality. (3)
c) Without carrying out a statistical test, conclude the acceptance or rejection of the null hypothesis H0 : p= 0 against the alternative Hi : p 0 at 5% level of significance.
(1)
[5]
Q 14) In an examination in Mathematics 20 randomly selected students at a government school had obtained mean mark of 52 and the sum of squares of deviations from this mean mark was 1312. From a private school, 17 randomly selected students, taking the same examination, obtained a mean mark of 36, and the sum of squares of deviations from this mean was 1401.
Assume that the marks obtained in government schools follow a normal distribution (, jf)and that in private schools follows a normal distribution (2, <j\ ), and that the two populations are independent.
a) Test the null hypothesis H0 : H = 2u2 vs H1:u1 2u2, when o\ =70 and a?, =90. (3)
b) Test the hypothesis H0 : ax2 = (2)
c) If your conclusions is to accept the hypothesis in (b), test the null hypothesis
H0 : h/h = 1 vs H1 : 1 when the common population variance is unknown. Comment on testing H0, if the hypothesis in (b) is rejected. (3)
d) Assuming the population variances to be equal, obtain 95% confidence interval
Q 15) In a spring balance, when weights are added to the scale pan, the spring stretches. The following table shows the results obtained when different load X (in Newtons) were applied in a random order and the length of the spring Y (in cms).
X : 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Y : 10.7 11.3 12.0 12.4 13.0 13.7 14.5 15.1 15.6 16.0
a) Representing the relationship between X and Y as Yi = a + pXi + ei
where eis are iid N(0, a2) random variables, estimate a and P by the method of least squares.
(5)
(1)
(2)
(4)
[12]
b) If the problem were to be to estimate a and P by maximum likelihood (ML) method, what will be your answer?
c) Estimate the value of a2.
d) Obtain the residuals using the fitted regression and plot the same against the fitted responses (Y ).Comment on the adequacy of the model
Q 16) A company employs drivers to work round the clock. The drivers union is concerned that some periods are more dangerous than to others. The management contests this, claiming that no one period is more dangerous than any other. The following data gives the details of the incidence of traffic accidents by time of day for this group over the previous two years.
|
Time of Day |
No. of Accidents |
|
00.01 - 04.00 |
14 |
|
04.01 - 08.00 |
16 |
|
08.01 - 12.00 |
24 |
|
12.01 - 16.00 |
22 |
|
16.01 - 20.00 |
24 |
|
20.01 - 24.00 |
20 |
a) Are these results consistent with the management claim? (4)
b) The company wishes to examine whether there is an association between accidents proneness and colour blindness. The results for a group of 80 drivers (with a minimum of 5 years employment) are as in the following table.
| ||||||||||||
|
Is there sufficient evidence to conclude that there is an association between colour blindness and accident proneness at 5% level? (3) | ||||||||||||
INSTITUTE OF ACTUARIES OF INDIA
EXAMINATIONS 20th October 2009 Subject CT3 - Probability & Mathematical Statistics Time allowed: Three Hours (15.00 - 18.00 Hrs)
Total Marks: 100
INSTRUCTIONS TO THE CANDIDATES
1. Please read the instructions on the front page of answer booklet and instructions to examinees sent along with hall ticket carefully and follow without exception
2. Mark allocations are shown in brackets.
3. Attempt all questions, beginning your answer to each question on a separate sheet. However, answers to objective type questions could be written on the same sheet.
4. In addition to this paper you will be provided with graph paper, if required.
AT THE END OF THE EXAMINATION Please return your answer book and this question paper to the supervisor separately.
Q. 1) A sample of 280 observations has mean 54 and standard deviation 3. It was found later that two observations which should read as 62 and 82 had been wrongly recorded as 64 and 80 respectively.
Calculate the correct mean and standard deviation. [3]
Q. 2) Find the probability that a leap year selected at random will contain 53 Sundays. [2]
Q.3) A random variable X has a mixed distribution with probability function P (X = 0) = a;
fx(x) = bx2 (1 x), 0 < x < 1
0, otherwise
If mean of X is "a", find a and b, and hence, find the standard deviation of X. [3]
Q.4) If G(t) is the Probability Generating Function of a random Variable X and a>0, then show that
[2]
0
Q. 5) Let X and Y be jointly distributed with probability density function,
f(x _ j ~A1 + xy)> l*l<lyK1
0, otherwise
Show that X and Y are not independent but X2 and Y2 are independent [5]
Q. 6) If the random variable Xfollows Poisson distribution with parameter A , then show that E (\X- 1|)= A - 1 + 2 e-
[4]
Q. 7) A random variable X has the following probability function: r c
P (x) = jx(x+l)' X 123-
I 0, otherwise
(a) Find the value of C. (1)
(b) Show that E (X) does not exist (1)
(c) If (t) (for t < 0) is the Moment Generating Function then, prove that
(1 _ et) = eCM*- *) (4)
An executive (HR) of a life insurance company desires to analyse the salaries of 27 actuarial staff employed by his company. The Manager (HR) advised the executive that salary should be based on number of actuarial papers cleared and years of professional experience in life insurance industry.
Q.8)
Suggest a suitable statistical model to quantify, defining the symbols used. [3]
Let X1,X2,.........,Xnbea random sample with common pdf
Q.9)
f(x) = f1' < X < 1 J 10, otherwise
If U = Min (X1,X2,.........,Xn) and V = Max (X,X2,.........,Xn)
(a) Find the pdf of U (2)
(b) Find mean and variance of U (3)
(c) Find the pdf of V (1)
(d) Show that
E (V) = n E (U)
Var (V) = Var (U) (2)
(a) Define Critical Region of a test? (1)
Q.10)
(b) Let p denote the probability of getting a head when a given coin is tossed once. Suppose that the hypothesis H0: p = 0.5 is rejected in favor of H- p = 0.6, if 10 trials results in 7 or more heads.
Calculate the size and power of the test (4)
Two independent normal random variables X and Y have means |j. and p respectively, and both have variance 1.
(a) Show that the distribution of random variable Y- pX depends on p, but not on |j. (2)
Q.11)
(b) A single observation is made on X and Y, resulting in values x and y respectively.
Use your result to test the hypothesis p = 1 against the alternative p * 1 at 5% level (2) when x = 0.3 and y = 2.7
(c) Comment on the result of (b) (1)
[5]
Q.12)
Let Y be the number of patients dying in a region among x patients tested positive H1N1 virus, is to be modelled as a Poisson random variable with mean 0x, where 9 is unknown.
Suppose data is available from n independent regions: region i with Xi patients tested positive H1N1 virus of which have died, i = 1,2,3,.........,n.
The least squares estimator of 9 is that value of 9 for which f=i(Yi E(Y{)Y is minimized.
(a) Show that the least squares estimator of 0 is given by:
r.v.
All-1 Atil
s = (3)
(b) Find 9, the maximum likelihood estimator of 9. (3)
(c) Examine whether 9 and 9 provide unbiased estimators of 0. (2)
(d) The table below gives the region- wise patients affected by H1N1 virus and the number of patients died:
(3)
[11]
Page 4 of 7
|
Region |
Number of patients affected by H1N1 virus |
Number of deaths |
|
1 |
41,773 |
534 |
|
2 |
77,578 |
1,025 |
|
3 |
1,205 |
2 |
|
4 |
22,422 |
74 |
|
5 |
9,965 |
97 |
|
6 |
29,223 |
67 |
Find the value of 9 and 9 from the given data.
Q.13) Two catalysts are being analysed to determine how they affect the mean yield of a chemical process. Specifically, catalyst 1 is currently in use, but catalyst 2 is acceptable. Since catalyst 2 is cheaper, it should be adopted, provided it does not change the process yield.
A test is run in the pilot plant and results are shown in the following table:
|
Observation number |
Catalyst 1 (x) |
Catalyst 2 (y) |
|
1 |
91.50 |
89.19 |
|
2 |
94.18 |
90.95 |
|
3 |
92.18 |
90.46 |
|
4 |
95.39 |
93.21 |
|
5 |
91.79 |
97.19 |
|
6 |
89.07 |
97.04 |
|
7 |
94.72 |
91.07 |
|
8 |
89.21 |
92.75 |
(a) Test at 5% level with clearly mentioning the hypothesis, is there any difference
between the mean yields, if:
(i) Variance of x (ct*) = 8.04 and Variance of y (tfy) = 14.84 (4)
(ii) Equal variance and unknown (5)
(b) Test whether the variances are equal at 5% level (2)
(c) Find 95% confidence interval for the difference of mean yields using result in (b). (2)
(d) Comment on the results as obtained in (c) (1)
[14]
Q.14) For a particular week, the following table gives minimum temperature (X), maximum temperature (Y) on the weekdays and maximum temperature (Z) on the week-end recorded in 11 Indian cities. (The temperatures are in Degree Celsius).
|
City |
Minimum Temperature (x) |
Maximum Temperature on a Weekday (y) |
Maximum Temperature on a Weekend (z) |
|
1 |
10.00 |
22.40 |
20.70 |
|
2 |
8.00 |
21.84 |
20.43 |
|
3 |
13.00 |
21.35 |
25.31 |
|
4 |
9.00 |
22.15 |
20.56 |
|
5 |
11.00 |
22.27 |
20.80 |
|
6 |
14.00 |
20.43 |
21.19 |
|
7 |
6.00 |
20.25 |
20.21 |
|
8 |
4.00 |
17.66 |
19.90 |
|
9 |
12.00 |
21.90 |
20.95 |
|
10 |
7.00 |
21.15 |
20.31 |
|
11 |
5.00 |
19.05 |
20.09 |
Y.x = 99.00; Y.X2 = 1,001.00; y = 230.45; Ey- 4,850.30; = 2,104.85
(a) Draw a scatter plot of y against x, and comment on whether a line of regression is appropriate to model the relationship between minimum temperature and maximum temperature on weekday (2)
(b) Draw the scatter plot of z against x, and comment on your plot (2)
(c) Fit a least squares regression line of y on x and calculate its parameters. (3)
(d) Show that the least square fitted regression equation of z on x is same regression line
as obtained in above (c) (3)
(e) Calculate the correlation coefficient of
(i) x and y,
(ii) x and z,
Comment on result (2)
(f) It was later found that the noting of the values for city 3 were wrong, so decided to omit those values.
(i) Calculate the regression line of z on x (5)
(ii) Calculate the correlation coefficient of x and z (1)
(g) Comment briefly on your result as obtained in above (f)(ii) (1)
Q.15) (a) State the mathematical model of one-way ANOVA defining all notations and
assumptions (2)
(b) Random sample of claim amounts (in units if Rs.1,000 over a total 3-year period) under hospitalization reimbursement policies were taken from 5 different general insurance companies are shown below.
|
Company | ||||
|
y |
72 |
ys |
y4 |
ys |
|
60 |
40 |
33 |
55 |
88 |
|
52 |
56 |
0 |
78 |
89 |
|
11 |
123 |
0 |
99 |
44 |
|
33 |
0 |
12 |
234 |
78 |
|
1 |
12 |
19 |
45 |
85 |
|
87 |
54 |
23 |
67 | |
|
110 |
77 |
67 | ||
|
24 | ||||
yi = 354; y2 = 386; y3 = 87 ; y4 = 645 ; y5 = 384
Ey? = 27,184 ; yf = 29,430 ; yf = 2,123 ; y| = 84,669 ; Eyf = 30,910
Test for the equality of mean claim amounts of the five companies, using ANOVA. (4)
(c) A marketing analyst claims that salaries of actuarial students do not depend on number of actuarial papers they have cleared. To test his claim, he collects data of 158 actuarial students distributed according to their annual salaries and number of papers cleared as shown below
|
Papers cleared |
(A c la (A | |||
|
3 - 5 |
5 - 8 |
8 - 10 |
10 - 12 | |
|
0 - 3 |
45 |
20 |
6 |
5 |
|
i 6 |
7 |
20 |
9 |
6 |
|
1 i 9 |
5 |
8 |
15 |
12 |
Conclude your view statistically based on the above data using a x2 test (4)
Page 7 of 7
|
Attachment: |
| Earning: Approval pending. |
