ST2352 2012 Problem Sheet 2: Revision of Binomial/Poisson distributions
The following is a table of numbers Z drawn randomly from U(0,1)
0.594
0.817
0.090
0.067
0.820
0.133
0.726
0.270
0.844
0.546
0.472
1
0.051
0.802
0.094
0.268
0.896
0.235
0.302
0.256
0.982
0.473
0.212
0.156
0.696
0.088
0.666
0.855
0.055
0.565
0.124
0.931
0.213
0.174
0.968
0.572
0.597
0.508
0.332
0.985
0.520
0.703
0.460
0.235
0.346
0.220
0.034
0.329
0.842
0.704
0.342
0.558
0.534
0.374
0.096
0.420
0.203
0.786
0.778
0.612
0.182
0.506
0.749
0.863
0.462
0.783
0.697
0.539
0.496
0.142
0.197
0.558
0.980
0.171
0.842
0.197
0.089
0.173
0.960
0.082
0.384
0.113
0.139
0.715
0.315
0.331
0.044
0.570
0.691
0.262
0.085
0.248
0.835
0.027
0.917
0.348
0.075
0.592
0.750
0.452
0.000
0.434
0.489
0.245
0.687
0.268
0.692
0.679
0.522
0.787
0.897
Use these numbers to sample 10 replications of the ‘experiment’ : roll a die 6 times and count the number N
of 6’s in these 10. Render the results as a relative frequency table. Compute – both from the replications and
from the table – the mean and the variance of the realised values of N. What is the probability distribution of
2
N? What are its expected value and variance? Contrast

1
 xi  x  
 For the moment use the sample variance formula n


these with your arithmetical samples.

2
Consider now the data in Q1 as two sets of 10 replications of rolling a dice 3 times. Denote the counts as N1
and N2. Note N1 +N2 = N. What are the average and variance of the sampled values of N1 and N2. What are
the separate probability distributions of N1 and N2? What are their expected values and variances? Compute
the joint probability distribution of N1 and N2. Use this to derive the probability distribution of N1 + N2.
3
Write down the probability generating functions for N1 and N2 and N1 + N2. Write down the moment
generating functions for N1 and N2 and N1 + N2.
4
“Roll the die” n=20 times; note by I = 0 and I = 1 the events ‘”not 6” and “6”. This is a Bernoulli sequence.
What is the (theoretical; n=∞) probability distribution of the number of instances of “0” between successive
values of “1”. Suppose the most recent value of I is I = 0. What is the prob distribution of the number of
instances of “0” until the next “1”? Compute both the pmf and the cdf.
5
In each of many intervals of length  an event occurs (I = 1 ) with probability . What are the pmf and the
cdf of the total time until the next event? (time = , 2….) Discuss the case of  0, when the time T to the
next event is real valued random variable with and Exponential distribution.
6
If T follows an exponential distribution with mean , realisations may be sampled by T = -  ln(1-Z), using
values of Z in the tables above. Use arguments based on the event identity below to justify this. With  = 2,
simulate 20 realisations of inter event times T. For your simulation how many events are there in each of the
the intervals (0,1), (1,2), (2,3) …? Denote by Y1 the random
Identity (T  t )     ln(1  Z )  t   ln(1  Z )  t  ..

variable “number of events in unit time”. What is the
probability distribution of Y1?

7

If Y1 denoted the number of events in the first period and Y2 that in the second what is the distribution of
Y1+ Y2? Explain using mgf arguments. If T1 denotes the time to first event (exponential dist) and T2 the
additional time to the second what is the distribution of T1+ T2? Use mgf
(Time to 2nd  t )  (Num events in t is <2)
arguments to show that the distribution is not exponential. Use the
event identity to derive the cdf and hence the pdf of T1+ T2.