THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
Semester 1 2016
MATH2801
Theory of Statistics
TIME ALLOWED -Two (2) hours
TOTAL NUMBER OF QUESTIONS - 4
ANSWER ALL QUESTIONS
THE QUESTIONS ARE OF EQUAL VALUE
ANSWER EACH QUESTION IN A SEPARATE BOOK
THIS PAPER MAY BE RETAINED BY THE CANDIDATE
ONLY CALCULATORS WITH AN AFFIXED "UNSW APPROVED"
STICKER MAY BE USED
Statistical tables are provided at the end of the examination paper.
All answers must be written in ink. Except where they are expressly required pencils
may only be used for drawing, sketching or graphical work.
Semester 1 2016 MATH2801 Page 2
1. [20 marks] Answer this question in a separate book
a) Discrete random variables X and Y have the following joint distribution:
fx,Y(x, y)
y
0
0 4/16
X l 4/16
1
4/16
3/16
2
1/16
0
i) Are X and Y independent? Give a reason for your answer.
ii) Consider the product XY.
A) Calculate the probability P(XY > 0).
B) Compute E(XY) and Var(XY).
b) Let X and Y be continuous random variables with joint density:
3 fx,y(x,y) = 2(x
2 + y2 ) for 0 < x < l, 0 < y < l.
i) Sketch clearly the region of positive density.
ii) Compute the marginal density function fy(y).
iii) A) Compute the conditional density fx1Y=ixly).
B) Hence determine E(XIY = y).
C) Hence determine E(XIY = 0.6).
iv) Are X and Y independent? Give a reason for your answer.
Please see over ...
Semester 1 2016 MATH2801 Page 3
2. [20 marks] Answer this question in a separate book
a) An insurance company insures an equal number of male and female
clients. In any given year the probability that a randomly chosen male
client makes a claim is p1 , and the probability that a randomly chosen
female client makes a claim is p2 .
Each year a client is chosen at random for a policy review, and the client
chosen in any year is independent of the choice in any other year.
i) In any year, what is the probability that the randomly chosen client
made a claim on their policy in the previous year.
ii) What is the probability that, in two successive years, each of the
randomly chosen clients made a claim in the year prior to the time
in which he/she is selected.
iii) Suppose that for two consecutive years each of the randomly chosen
clients had made a claim in the year prior to the time in which
he/she is selected. What is the conditional probability that both of
the clients are female?
b) Victor and Wilfred arrive at a railway station independently of one an-
other at a random time between 1pm and 2pm. Let V be the number
of minutes past 1pm when Victor arrives, and let W be the number of
minutes past 1pm when Wilfred arrives.
The marginal density functions of V and W are both uniform:
1 fv(v) = fw(w) = 60 , 0 '.S V < 60, 0 '.S W < 60.
i) Briefly explain ( one sentence only is required) why the joint density
function of arrival times of Victor and Wilfred is
1 fv,w( v, w) = 3600' 0 '.S V < 60, 0 '.S W < 60.
ii) Using fv,w(v, w), or otherwise, determine the probability that Victor
arrives before Wilfred, i.e find P(V < W).
iii) Let T = W - V be the time (in minutes) between Victor's arrival
and Wilfred's arrival.
A) Determine the cumulative distribution function Fy(t).
B) Hence, or otherwise, determine the density function jy(t).
Please see over ...
Semester 1 2016 MATH2801 Page 4
3. [20 marks] Answer this question in a separate book
a) Let X be a binomial variable, X rv Bin(n,p) for some 0 < p < 1, so
P( X = x) = f x ( x) = (:) PX (1 - p r-x for x = 0, 1, 2, ... , n
with JE(X) = np and Var(X) = np(l - p).
i) Prove that X has moment generating function
ii) Let Y = 2X.
A) Determine my(u), the moment generating function for Y.
B) Determine fy(y), the probability function for Y.
C) Does Y have a binomial distribution for some suitable value of
the parameters? Give a reason for your answer.
b) A random sample of 100 postal employees found that the average time
the employees had worked at the postal service was ?f = 7 years with a
(sample) standard deviation of s = 2 years. Does this provide convincing
evidence that the mean time of employment (in years) has changed from
the value of 7.5 that was true 20 years ago?
Carry out an appropriate hypothesis test to answer this question by an-
swering the following:
i) State the null and alternative hypotheses.
ii) State any assumption(s) you are making in order to carry out this
hypothesis test.
State what you could do, if anything, to check the plausibility of your
assumption( s).
iii) State the formula for the test statistic and the null distribution.
iv) Compute the observed value of the test statistic.
v) Give an expression for the P-value, and then estimate it as best you
can from the tables provided.
vi) Write a relevant conclusion concerning the mean time of employment
of postal employees.
Please see over ...
Semester 1 2016 MATH2801 Page 5
4. [20 marks] Answer this question in a separate book
We wish to estimate p, the chance of obtaining a head when a particular coin
is flipped. To do so, the coin is randomly flipped many times.
For i = 1, ... , let Xi = 1 if a head is obtained on the ith flip and Xi = 0
otherwise. The Xi are i.i.d. Bernoulli(p) random variables for some 0 < p < l
and the probability function can be given by
P(Xi = x) = f(x) = {P
1-p
or equivalently by
for X = l
for X = 0
for X = 0, 1.
Suppose the coin is to be flipped n times, resulting in observations x1, x2, . .. , Xn,
The sample proportion of heads can be computed by p = I::~o Xi = x.
n
a) Determine the method of moments estimator of p. Show that it is
given by the sample proportion p.
b) Determine the maximum likelihood estimator of p. Show that it is
given by the sample proportion p.
c) Determine the Fisher information for p.
d) i) Write down the large sample approximation to the distribution of p.
ii) Use the delta method to deduce the large sample approximation to
the distribution of 1 fJ .. -p
e) Assume now that the coin is flipped n = 100 times, and that a head is
obtained 75 times in the 100 flips of the coin. Compute an (approximate)
95% confidence interval for p.
Semester 1 2016 MATH2801 Page 6
Table of some common distributions
X mass/ density domain mean variance mgf
function mx(u) = E(euX)
Bernoulli P(X = 1) = p {0,1} p pq q+peu
P(X = 0) = q = 1 - p
Binomial
Bin(n,p)
G)Pk(l - p)n-k k = 0,1, ... ,n np np(l - p) (1-p+peur
Geometric p(l-p)k-1 k = l,2, ... l (1-p) p p ~ e-"-l+p
Poisson e->->.k /kl k = 0,1, ... ,\ ,\ exp{,\(eu -1)}
Uniform (b - a)-1 xE(a,b) ½(a+b) f2(b-a) 2 ebu_eau u(b-a)
Exponential 1 _],_ X xE(0,oo) f3 (32 1 73e f3 1-,Bu
Normal
N(µ,a-2)
_l_ exp { - (x-µ)2 }
v2-1ra2 ~ x E {-oo, oo} µ (T2 eµu+½CT2u2
Gamma (a,/3) e-xff3xa.-l xE(0,oo) Ol/3 a/32 ( 1-1/lu r r(a)/l"
Beta /3( a, /3) r(a+/l) a-1c1- )/l-1 XE [0,1) "' a,B r(a)r(,B)x x a+,B (a+,B)2(a+/J+l)
Some formulae
Let X1, ... , Xn be independent N(µ, cr2) variables. Then X and S2 are independent,
and
(n-l)S2 2
---- ,.___, Xn-1 ·
cr2
The Delta Method
Let 01 , 02 , ... be a sequence of estimators of 0 such that
en -0 v
cr/yn,-+ N(O, 1).
Suppose the function g is differentiable at 0 and g' ( 0) -/= 0. Then
g(0n) - g(0) !!._,, N(O, 1).
g'(0)cr I vn
Semester 1 2016 MATH2801 Page 7
t distribution critical values
Key: Table entry for p and C is the critical value t* with probability p lying to its right
and probability C lying between -t* and t*.
Upper tail probability p
df .25 .20 .15 .10 .05 .025 .02 .01 .005 .0025 .001
1 1.000 1.376 1.963 3.078 6.314 12.71 15.89 31.82 63.66 127.3 318.3
2 0.816 1.061 1.386 1.886 2.920 4.303 4.849 6.965 9.925 14.09 22.33
3 0.765 0.978 1.250 1.638 2.353 3.182 3.482 4.541 5.841 7.453 10.21
4 0.741 0.941 1.190 1.533 2.132 2.776 2.999 3.747 4.604 5.598 7.173
5 0.727 0.920 1.156 1.476 2.015 2.571 2.757 3.365 4.032 4.773 5.893
6 0.718 0.906 1.134 1.440 1.943 2.447 2.612 3.143 3.707 4.317 5.208
7 0.711 0.896 1.119 1.415 1.895 2.365 2.517 2.998 3.499 4.029 4.785
8 0.706 0.889 1.108 1.397 1.860 2.306 2.449 2.896 3.355 3.833 4.501
9 0.703 0.883 1.100 1.383 1.833 2.262 2.398 2.821 3.250 3.690 4.297
10 0.700 0.879 1.093 1.372 1.812 2.228 2.359 2.764 3.169 3.581 4.144
11 0.697 0.876 1.088 1.363 1.796 2.201 2.328 2.718 3.106 3.497 4.025
12 0.695 0.873 1.083 1.356 1.782 2.179 2.303 2.681 3.055 3.428 3.930
13 0.694 0.870 1.079 1.350 1.771 2.160 2.282 2.650 3.012 3.372 3.852
14 0.692 0.868 1.076 1.345 1.761 2.145 2.264 2.624 2.977 3.326 3.787
15 0.691 0.866 1.074 1.341 1.753 2.131 2.249 2.602 2.947 3.286 3.733
16 0.690 0.865 1.071 1.337 1.746 2.120 2.235 2.583 2.921 3.252 3.686
17 0.689 0.863 1.069 1.333 1.740 2.110 2.224 2.567 2.898 3.222 3.646
18 0.688 0.862 1.067 1.330 1.734 2.101 2.214 2.552 2.878 3.197 3.610
19 0.688 0.861 1.066 1.328 1.729 2.093 2.205 2.539 2.861 3.174 3.579
20 0.687 0.860 1.064 1.325 1.725 2.086 2.197 2.528 2.845 3.153 3.552
21 0.686 0.859 1.063 1.323 1.721 2.080 2.189 2.518 2.831 3.135 3.527
22 0.686 0.858 1.061 1.321 1.717 2.074 2.183 2.508 2.819 3.119 3.505
23 0.685 0.858 1.060 1.319 1.714 2.069 2.177 2.500 2.807 3.104 3.485
24 0.685 0.857 1.059 1.318 1.711 2.064 2.172 2.492 2.797 3.091 3.467
25 0.684 0.856 1.058 1.316 1.708 2.060 2.167 2.485 2.787 3.078 3.450
26 0.684 0.856 1.058 1.315 1.706 2.056 2.162 2.479 2.779 3.067 3.435
27 0.684 0.855 1.057 1.314 1.703 2.052 2.158 2.473 2.771 3.057 3.421
28 0.683 0.855 1.056 1.313 1.701 2.048 2.154 2.467 2.763 3.047 3.408
29 0.683 0.854 1.055 1.311 1.699 2.045 2.150 2.462 2.756 3.038 3.396
30 0.683 0.854 1.055 1.310 1.697 2.042 2.147 2.457 2.750 3.030 3.385
40 0.681 0.851 1.050 1.303 1.684 2.021 2.123 2.423 2.704 2.971 3.307
50 0.679 0.849 1.047 1.299 1.676 2.009 2.109 2.403 2.678 2.937 3.261
60 0.679 0.848 1.045 1.296 1.671 2.000 2.099 2.390 2.660 2.915 3.232
80 0.678 0.846 1.043 1.292 1.664 1.990 2.088 2.374 2.639 2.887 3.195
100 0.677 0.845 1.042 1.290 1.660 1.984 2.081 2.364 2.626 2.871 3.174
1000 0.675 0.842 1.037 1.282 1.646 1.962 2.056 2.330 2.581 2.813 3.098
.50 .60 .70 0.80 .90 .95 .96 .98 .99 .995 .998
Probability C
.0005
636.6
31.60
12.92
8.610
6.869
5.959
5.408
5.041
4.781
4.587
4.437
4.318
4.221
4.140
4.073
4.015
3.965
3.922
3.883
3.850
3.819
3.792
3.768
3.745
3.725
3.707
3 .. 690
3.674
3.659
3.646
3.551
3.496
3.460
3.416
3.390
3.300
.999
Semester 1 2016 MATH2801 Page 8
Standard normal probabilities
Key: Table entry for z is the area under the standard normal curve to the left of z .
z . 00 .01 .02 .03 .04 .05 .06 .07 .08 .09
-2.9 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.0014
-2.8 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.0019
-2.7 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.0026
-2.6 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.0036
-2.5 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.0048
-2.4 0.0082 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 0.0064
-2.3 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.0084
-2.2 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0113 0.0110
-2.1 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.0143
-2.0 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183
-1.9 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.0233
-1.8 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.0301 0.0294
-1.7 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.0367
-1.6 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455
-1.5 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.0559
-1.4 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.0681
-1.3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823
-1.2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985
-1.1 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.1170
-1.0 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.1379
-0.9 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.1611
-0.8 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.1867
-0.7 0.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.2148
-0.6 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451
-0.5 0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.2776
-0.4 0.3446 0.3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.3156 0.3121
-0.3 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520 0.3483
-0.2 0.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3897 0.3859
-0.1 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.4247
-0.0 0.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.4641
0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359
0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753
0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141
0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517
0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879
0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224
0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549
0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852
0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133
0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389
1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621
1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830
1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015
1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177
1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319
1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441
1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545
1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633
1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706
1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767
2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817
2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857
2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890
2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916
2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936
2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952
2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964
2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974
2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981
2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986