Paper B STA02A2 2023

Question 1 (2 marks)
You invited 7 friends for a party at your house: 3 are friends from high school, and 5 are friends from
university. All 7 line up at random in a row for a photo (you are not in the photo as you are taking the photo).
How many different line up’s are possible such that your friends from high school stand next to each other,
and your friends from university stand next to each other?

Question 2 (5 marks)
At a small insurance company, 2 clerks process claims by completing an electronic form. Clerk A processes
60% of all claims, and Clerk B processes the rest. The probability that Clerk A makes an error is 0.04, and the
probability that Clerk B makes an error is 0.1.
2.1 (1 mark) What is the probability that a claim is processed by Clerk A and it contains an error?
2.2 (2 marks) What percentage of claims are processed correctly, i.e., without any errors?
2.3 (2 marks) What is the probability that a randomly selected claim that contains an error, was processed
by Clerk B?

Question 3 (3 marks)
Consider two non-zero events A and B such that P ( A  B ) = P ( B ) . Use the law of total probability to show

that A and B are mutually exclusive, and state, with a valid reason, whether A and B can be independent.

Question 4 (6 marks)
4.1 (2 marks) Let U be uniform on [0, 1], and let X = F −1 (U ) . Show that the cdf of X is F.

4.2 (4 marks) The cumulative distribution function of a random variable X is:

1 − e−2 x x0
F ( x) = 
0 otherwise

Find the probability density function, f ( y ) , and cumulative distribution function, F ( y ) , for the

random variable Y = e X . Specify the valid values of Y for both the probability density function and the
cumulative distribution function.

Question 5 (3 marks)
A coin is biased such that the probability of it landing on TAILS is 0.75. In a game of chance, the coin is
tossed until the 4th time that it landed on TAILS.
5.1 (1 mark) Define this random variable and its parameters.
5.2 (2 marks) What is the probability that the coin must be tossed 10 times to get 4 TAILS?

1
Paper B STA02A2 2023

Question 6 (5 marks)
Consider the following cumulative distribution function of a random variable X.
0 x0
1 2
F ( x) =  8 ( x + 2x ) 0  x  2
 x2
1

6.1 (2 marks) Use the cdf to calculate P (1  X  3) .

6.2 (3 marks) Derive the probability density function of X and calculate E ( X 2 ) .

Question 7 (4 marks)
The joint density distribution of variables X and Y is f ( x, y ) , for 0  x  1 and y  1 . Draw the valid region

for the sample space and add the valid region for the event space to calculate P (Y  2 X ) from such a joint

density function. Give the coordinates of the valid region. Do not attempt to calculate a probability, just draw
the sample space and event space in one graph, together with the coordinates.

Question 8 (12 marks)

Consider the following joint density distribution of variables X and Y:
6− x− y
f ( x, y ) = for 0  x  2 and 2  y  4
8

8.1 (4 marks) Find the equation of the conditional density f ( x | y ) .

8.2 (3 marks) Calculate P ( X  1.5 | Y = 2.2 ) .

8.3 (5 marks) The E ( X ) = 0.833 and E (Y ) = 2.833 . Calculate the covariance between X and Y.

Question 9 (11 marks)

9.1 (2 marks) Show that, if random variables X and Y are independent, then F ( x, y ) = F ( x ) F ( y ) .

9.2 (2 marks) A random variable X ~ gamma(α = 10, λ = 8,). If Y = 5 + 8 X , calculate the variance of Y.
9.3 (2 marks) Variables X and Y are independent random variables with moment-generating functions M X

and M Y , and Z = X + Y . Show that M Z ( t ) = M X ( t ) M Y ( t ) .

−1
 t
9.4 (5 marks) The moment-generating function of X ~ exponential(λ) is M X ( t ) = 1 −  . Use this M X ( t )
 
to find the variance of X.

2
Paper B STA02A2 2023

Question 10 (5 marks)
In a laboratory, a chemical reaction occurs in two stages. The reaction time in the first stage (U) is uniformly
distributed over the interval [1,3]. The reaction time of the first stage has an effect on the reaction time in the
second stage (T). Therefore, the reaction time of the second stage depends on the reaction time in the first
stage, such that T | U ~ exponential(u). Find the expected reaction time in the second stage.

Question 11 (7 marks)
A certain charity receives donations from the public. The amounts that people donate are independently and
identically distributed random variables, X i , with a mean of R50 and a standard deviation of R25.

11.1 (3 marks) State the weak law of large numbers.

11.2 (4 marks) The charity needs between R5000 and R5500 to complete a project. Use the central limit
theorem to calculate the probability (to 4 decimal places) that donations from the next n = 100 people
will be enough for the charity to complete the project.

Question 12 (7 marks)
n
12.1 (2 marks) For Z ~ N ( 0,1) , what is the distribution of U =  Z i2 ?
i =1

12.2 (5 marks) X 1 , X 2 , , X n are independent and identically distributed observations of a random sample

of size n, where:

X i ~ N ( , 2 )
1 n 1 n
 Xi  ( Xi − X )
2
X= S2 =
n i =1 n − 1 i =1

X −
U = Z
n
Z ( n − 1) S 2
Z= ~ N ( 0,1) 2
V= ~ tn W= ~  n2−1
 i =1
i
U 2

n
n

Use your answer in Question 12.1 and the results given above to show that:
X −
T= ~ tn −1
S
n

3
Paper B STA02A2 2023

Standard Normal Distribution

Entry represents area under the cumulative standard normal distribution from −∞ to z
Z 0
z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
-3.4 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0002
-3.3 0.0005 0.0005 0.0005 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0003
-3.2 0.0007 0.0007 0.0006 0.0006 0.0006 0.0006 0.0006 0.0005 0.0005 0.0005
-3.1 0.0010 0.0009 0.0009 0.0009 0.0008 0.0008 0.0008 0.0008 0.0007 0.0007
-3.0 0.0013 0.0013 0.0013 0.0012 0.0012 0.0011 0.0011 0.0011 0.0010 0.0010
-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

4
Paper B STA02A2 2023

Standard Normal Distribution (continued)

Entry represents area under the cumulative standard normal distribution from −∞ to z
0 Z
z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
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
3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990
3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993
3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995
3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997
3.4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998

5
Paper B STA02A2 2023

FORMULA SHEET
Discrete
Probability Mass Function Valid Range E(X) Var(X)
Distributions
n
p ( x ) =   p x (1 − p ) np (1 − p )
n− x
Binomial x = 0,1, n np
 x
1 1− p
p ( x ) = (1 − p )
x −1
Geometric p x = 1, 2,
p p2
 x − 1 r r = 1, 2, r (1 − p ) r (1 − p )
p ( x) =   p (1 − p )
Negative x−r

binomial  r −1  x = r , r + 1, p p2
 r  n − r 
   m = 1, 2, ,n
 x  m − x 
Hypergeometric p ( x) = r = 0,1, ,n NA NA
n 
  x = 0,1, ,m
m
e−   x
Poisson p ( x) = x = 0,1,  
x!

Continuous
Probability Density Function Valid Range E(X) Var(X)
Distributions
b+a (b − a )
2
1
Uniform f ( x) = a xb
b−a 2 12
x0 1 1
Exponential f ( x ) = e− x
 0  2
   −1 −  x x0  
Gamma f ( x) = x e
 ( ) ,  0  2
1  x− 
2
− 
Normal f ( x) =
1 
e 2   −  x    2
 2
 ( a + b ) a −1
f ( x) = x (1 − x )
b −1
Beta 0  x 1 NA NA
 ( a )  (b)

Properties of the Gamma Function


 ( ) =  x −1e− x dx  ( n ) = ( n − 1)! for n = 1, 2,3,
0

 ( + 1) =  ( )  ( 12 ) = 