The Open University Statistics - M248 TMA 01 Question Two

Complete the following paragraph by selecting words

You should be able to answer this question after working through Unit 2.
(a) Complete the following paragraph by selecting words or phrases from the list that follows it to fill in the underlined gaps.

In a long sequence of repetitions of a study or experiment, random samples tend to settle down towards probability distributions in the sense that, for discrete data, bar charts settle down towards probability functions and, for continuous data, histograms settle down towards probability functions. As the sample size increases, the amount of difference between successive graphical displays obtained from the data .

Available words and phrases: continuous cumulative decreases density discrete
frequency increases mass model models relative frequency remains constant unimodal unit-area [3]

(b) Kevin lives in a city which operates a bicycle hire scheme using a large number of bicycle ‘docking stations’ spread around the city. He walks past a small docking station, for up to six bicycles, each morning. Kevin has come up with the following probability mass function (p.m.f.) for the distribution of the random variable X which denotes the number of bicycles available at the docking station each morning.
It is given in
Table 1.
Table 1 The p.m.f. of X

x 0 1 2 3 4 5 6
p(x) 0.3 0.2 0.2 0.1 0.1 0.05 0.05
(i) What is the range of X? [1]
(ii) Explain why the p.m.f. suggested by Kevin is a valid p.m.f. [2]
(iii) What is the probability that, on any particular morning, there is one bicycle at the docking station? [1]
(iv) Write down a table containing values of F(x), the cumulative distribution function (c.d.f.) of X, for x = 0; 1; 2; 3; 4; 5; 6. [2]
(v) Write the probabilities P(X < 3) and P(X ≥ 5) in terms of the c.d.f. F(x). Use the c.d.f. to calculate the values of these two probabilities.

(c) In 1955, C.W. Topp and F.C. Leone introduced a number of
distributions in the context of the statistical modelling of the reliability of electronic components in engineering. One of these distributions has probability density function (p.d.f.) given by f(x) = 4x(1 − x)(2 − x) on the range 0 < x < 1.
(i) Verify, by integration, that Integrate( 4x(1 − x)(2 − x)) dx = x2(2 − x)2 + c; where c is an arbitrary constant

(ii) Explain why the p.d.f. suggested by Topp and Leone is a valid
p.d.f. [4]
(iii) What is the c.d.f. associated with this p.d.f.? [2] (iv) Suppose that X is a random variable following this p.d.f., and that we are interested in evaluating P(1/3 < X < 2/3). Write this probability in terms of the c.d.f., and hence show that P (1/3 < X < 2/3)= 39 81
(which is approximately 0.481)