Download MAS144 – Computational Mathematics and Statistics A (Statistics)

MAS1302 – Computational Probability and Statistics
Additional Exercises 2 (Not assessed)
1. Find the values of
(i) 8946 mod 4
(ii) 1067 mod 11
(iii) 124 mod 3
(iv) 999 mod 8
2. Consider the following congruential generator:
ri = (ari−1 + b) mod m, i = 1, 2, …, m.
Using a seed of 0, a multiplier of 4, an additive constant of 17 and a modulo of 27, generate the first 5
terms of the sequence produced by this generator. Is the maximum possible period for this choice of
modulo achieved? What is the period for this generator.
3. Suppose the outcome when a weighted six-sided dice is rolled is a discrete random variable X having
the following probability mass function:
x
Pr(X = x)
1
0.10
2
0.15
3
0.15
4
0.15
5
0.15
6
0.30
Suppose that the following is a list of random numbers u simulated from a U(0,1) distribution: 0.907,
0.792, 0.050, 0.125, 0.772, 0.343, 0.722, 0.572, 0.938, 0.860, 0.419, 0.215. For each
u, simulate an observation on the random variable X. If somebody had told you this was a fair dice,
would these outcomes have made you suspicious?
4. Let the discrete random variable X have a Geom(0.5) distribution. The following five numbers are
pseudo-random numbers from a U(0,1) distribution.
0.582, 0.339, 0.884, 0.004, 0.474
For each pseudo-random number u, simulate an observation on the random variable X.
5. Using the following set of eight random numbers from a U(0,1) distribution, simulate eight
observations on a Poisson(3) random variable X.
0.417,
0.486,
0.162,
0.108,
0.274,
0.882,
0.364,
0.119
[You can either use the formula for the Poisson distribution or use Neave tables. Work to four
decimals.]
6. Consult the table given out in lectures entitled “Random digits”. Starting on the right hand side, lower
half, take the first three digits to be the fractional part of a random observation u1 from a continuous
Uniform U(0,1) distribution. Then use successive groups of three digits for subsequent observations.
Thus, your u1 will be 0.467, your u2 will be 0.196, etc.
Let X ~ Bin(4, 0.6). Use your random numbers to generate two observations on X.