S2 Revision Quiz

Answers coming up…

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

Question 2 (3 marks)

X ~ R[0, 5]

Find E(X) and Var(X)

Question 3 (1 mark)

X ~ R[0, 5]

Find P(X > 3)

Question 4 (1 mark)

X ~ R[0, 5]

Find P(X = 3)

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

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.

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.

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.

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

Question 10 (1 mark)

A bag contains a large number of coins:75% are 10p coins25% are 5p coins

Write down the possible medians of all the samples of 3 coins that you could select from the bag

Question 11 (3 marks)

A bag contains a large number of coins:75% are 10p coins25% 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.

Question 12 (2 marks)

Write down the 2 conditions under which the Poisson distribution may be used as an approximation to the Binomial distribution.

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.

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.

Question 15 (3 marks)

A call centre routes telephone calls. The probability of a call being wrongly connected is 0.01The call centre receives 1000 calls each day. Find the mean and variance of the number of incorrectly connected calls.

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.

Question 17 (2 marks)

Write down 2 conditions for a Normal distribution to be used to approximate a Binomial distribution.

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.

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.

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.

Question 21 (3 marks)

Sketch the probability density function of X

Question 22 (1 mark)

What is the mode of X?

Question 23 (7 marks)

Specify fully the cumulative distribution function of X

Question 24 (3 marks)

Find the median of X using your answer for F(x)