1. S2 Revision Quiz Answers coming up…

2. Question 1 (3 points) A string AB of length 5 cm is cut, in a random place C, into two pieces. The random variable X is the length of AC. Write down the name of the probability distribution of X and sketch the graph of its probability density function

3. Question 2 (3 marks) X ~ R[0, 5] Find E(X) and Var(X)

4. Question 3 (1 mark) X ~ R[0, 5] Find P(X > 3)

5. Question 4 (1 mark) X ~ R[0, 5] Find P(X = 3)

6. Question 5 (1 mark) An engineering company makes an electrical parts. At the end of the process each part is checked to see if it is faulty. Faulty parts are detected at a rate of 1.5 per hour. Suggest a suitable model for the number of faulty parts per hour

7. Question 6 (2 marks) An engineering company makes an electrical parts. At the end of the process each part is checked to see if it is faulty. Faulty parts are detected at a rate of 1.5 per hour. Describe two assumptions that are necessary to model the number of faulty parts using a Poisson Distribution.

8. Question 7 (2 marks) An engineering company makes an electrical parts. At the end of the process each part is checked to see if it is faulty. Faulty parts are detected at a rate of 1.5 per hour. Find the probability of 2 faulty parts being detected in a 1 hour period.

9. Question 8 (3 marks) An engineering company makes an electrical parts. At the end of the process each part is checked to see if it is faulty. Faulty parts are detected at a rate of 1.5 per hour. Find the probability of at least one faulty part being detected in a 3 hour period.

10. Question 9 (3 marks) A bag contains a large number of coins: 75% are 10p coins 25% are 5p coins Write down all the possible combinations of 3 coins that you could select from the bag

11. Question 10 (1 mark) A bag contains a large number of coins: 75% are 10p coins 25% are 5p coins Write down the possible medians of all the samples of 3 coins that you could select from the bag

12. Question 11 (3 marks) A bag contains a large number of coins: 75% are 10p coins 25% are 5p coins A random sample of 3 coins is selected. Find the sampling distribution for the median of the values of the 3 selected coins.

13. Question 12 (2 marks) Write down the 2 conditions under which the Poisson distribution may be used as an approximation to the Binomial distribution.

14. Question 13 (2 marks) A call centre routes telephone calls. The probability of a call being wrongly connected is 0.01 Find the probability that 2 consecutive calls will be wrongly connected.

15. Question 14 (3 marks) A call centre routes telephone calls. The probability of a call being wrongly connected is 0.01 Find the probability that more than 1 in 5 consecutive calls will be wrongly connected.

16. Question 15 (3 marks) A call centre routes telephone calls. The probability of a call being wrongly connected is 0.01 The call centre receives 1000 calls each day. Find the mean and variance of the number of incorrectly connected calls.

17. Question 16 (2 marks) A call centre routes telephone calls. The probability of a call being wrongly connected is 0.01. The call centre receives 1000 calls each day. Use a Poisson approximation to find the probability that more than 6 calls each day are wrongly connected. Give your answer to 3 dp.

18. Question 17 (2 marks) Write down 2 conditions for a Normal distribution to be used to approximate a Binomial distribution.

19. Question 18 (2 marks) A Normal distribution is to be used to approximate a Binomial distribution. Write down the mean and variance of this normal approximation in terms of n and p.

20. Question 19 (5 marks) A factory makes 2000 DVDs each day. 3% of all DVDS made are faulty. Use a normal approximation to estimate the probability that at least 40 faulty DVDs are produced in a day.

21. Question 20 (3 marks) A factory makes 2000 DVDs each day. 3% are faulty It costs £0.70 to make each DVD. Non-faulty DVDs are sold for £11 each. Faulty DVDs are destroyed Find the expected profit made by the factory per day.

22. Question 21 (3 marks) Sketch the probability density function of X

23. Question 22 (1 mark) What is the mode of X?

24. Question 23 (7 marks) Specify fully the cumulative distribution function of X

25. Question 24 (3 marks) Find the median of X using your answer for F(x)