# Expert Work

Question 1
A random variable X has the following distribution:

x                            0                            5                            10
P(X = x)  0.4                        *                            0.3
(a). Is X discrete or continuous? [1 mark]

(b) Fill in the gap marked by *. [1 mark]

(c) State the concept on which your answer to part (b) is based. [1 mark]

(d) Calculate the mean and the variance of this distribution. [5 marks]
(e) Calculate P(X < 7) . [2 marks]

The local Automobile Association conducted a study of the incidences over a ten-year period of children under the age of 5 years who were involved in motor vehicle accidents that resulted in road fatalities. The results for children who were not wearing set belts at the time of the accident are shown in the following table:

Age                       Male             Female         Total
0                            265                227                492
1                            204                191                395
2                            367                207                574
3                            316                190                506
4                            477                194                671
Total                     1629              1009              2638

(a) Are these random variables discrete or continuous? [1 mark]

(b) What is the probability that a fatality involved a 3-year old? [2 marks]

(c) What is the probability that the fatality involved a child who is 3-year old and is a male? [2 marks]

(d) What is the probability that the fatality involved a child who is a 3-year old or is a male? [3 marks]
(e) What is the probability that the fatality involved a child who is 3-year old given that the child is male? [2 marks]

(f) Two fatalities are randomly selected, what is the probability that the first fatality involved a child who is less than 1 year old and the second fatality involved a child who is less than 1 year old? [3 marks]

Question 3

Consider the following question and the proposed solution:
The Quality Manager of a battery manufacturing plant reviewed the warranty records within his department and found that 4% of the low maintenance batteries produced at the plant over the past 5 years were returned within the warranty period for a confirmed defect. Twelve low maintenance batteries were purchased at the plant last Saturday. What is the probability that more than 2 of these batteries will be returned to the plant within the warranty period with a confirmed defect?

Let X be the number of batteries that were defective

P(X = 2) = 12C2(4%)10(96%)2 = 6.378-13

Critique the proposed solution. In your response, comment in: (1) the definition of the random variable X, (2) the suitability of the use of the binomial distribution, and (3) the working out of the answer.
State your answer to the question. Ensure that you define the random variable and show all the relevant working. [7 marks]