Download Tutorial sheet 5

17.1SB2 Statistics 2
Problem sheet 5 – Continuous distributions/Normal distribution
1.
Let X ~ Uniform(0, 1) and suppose that Y = g(X) = X3.
Identify the inverse mapping g-1 and the range of Y.
Using one of the formulae given in lectures (Part III , Section 5), calculate the pdf
of Y.
2.
Let X be a continuous random variable whose pdf is fX(x) = 2x, 0 < x < 1.
i) Sketch the graph of fX(x). Would you expect the mean of X to be greater or less
than 0.5?
ii) By evaluating appropriate integrals, calculate E(X), E(X2) and, hence, the
variance of X.
3.
Let X be a random variable whose distribution is N(100, 100).
Use tables of the Normal cumulative distribution function to calculate the
following probabilities:
i)
P(X > 115);
ii)
P(X < 92);
iii)
P(95 < X < 120).
(Hint: First transform X to a standard Normal random variable Z using the
transformation discussed in lectures.)
4.
The gestation period (measured in days) for human births can be taken as normal
N(, 2) with  = 266 and  = 16.
i) Calculate the probability that a gestation period lasts for more than 290 days.
ii) What is the probability of a gestation period lasting between 250 and 282
days?
5.
A medical trial was conducted to investigate whether a new drug extended the life
of patients with a certain disease. The drug was given to 38 patients and their
survival times (in months) were recorded. These were (in ascending order):
1
21
39
50
1
22
40
50
5
25
41
54
9
25
41
54
10
25
43
59
13
26
44
14
27
44
17
29
45
18
36
46
18
38
46
19
39
49
i)
Calculate the median and the quartiles Q1, and Q3 for these data.
ii)
Construct a stem-and-leaf diagram for the data. Does the plot support the
suggestion that the distribution of survival times is Normal?