Download B Normal distribution

A– LEVEL TOPIC REVIEW
unit S1
the Normal distribution
Give all answers correct to 3 significant figures where necessary.
1. The random variable Z is normally distributed with mean 0 and variance 1.
a) Sketch the probability function of Z, indicating the mean and the standard deviation on the zaxis.
(3 marks)
b) Find
P( Z  0.5)
(i)
(ii) P( Z  1.5)
(iii) P(1  Z  1)
(iv) E(2 Z  3)
(v) Var(1  3Z )
(5 marks)
2. The random variable X is normally distributed with mean 3 and variance 4.
a) Sketch the probability function of X, indicating the mean and the standard deviation on the xaxis.
(3 marks)
b) Express X in terms of Z, where Z is the random variable defined in Question 1.
(2 marks)
c) Find
P( X  2)
(i)
(ii) P( X  0)
(iii) P(1  X  5)
(iv) E(2 X  3)
(v) the standard deviation of 1  3X
(7 marks)
3. Given that X
N  4.5, 2.52  , find
a) P( X  4.2)
(2 marks)
b) P( X  5)
(2 marks)
c) P  X  4  1.5 
(3 marks)
4. The times taken by a group of people to run a marathon are modelled as a continuous random
variable having the normal distribution with mean 5 hours and standard deviation 1.5 hours.
a) Use this model to calculate
(i) the probability that a runner chosen at random took between 4 and 7 hours.
(3 marks)
(ii) the range, symmetrical about the mean, within which 85% of the runners’ times lie.
(4 marks)
b) Comment on the suitability of the normal distribution as a model in this case.
(2 marks)
5. The random variable X is distributed N   ,  2  .
Given that P( X  2)  0.125 and P( X  3)  0.271 , find the values of μ and σ.
(7 marks)
6. The random variable X is normally distributed with variance 900. The probability that X is greater
than 325 is 0.791, to 3 significant figures.
a) Calculate the mean of X.
(5 marks)
b) In forty independent observations of X, how many would you expect to be less than 325?
(2 marks)
A– LEVEL TOPIC REVIEW : ANSWERS
unit S1
the Normal distribution