MA22S6 Numerical and Data Analysis 1 2015-2016
Homework sheet 4
Due Thursday 25th of February
1. Explain the meaning of the expectation value and of the standard deviation. How
do you compute the expectation value and the standard deviation of a continuum
random variable given its probability density function? Same questions but given
the cumulative distribution function.
2. Suppose you are given the mean and variance for a continuous random variable X
which is
a) uniformly distributed;
b) exponentially distributed;
c) normally distributed;
Compute in each case the mean and variance for the continuous random variable
Y , related linearly to X as follows:
Y = 2X − 5
3. Suppose that the profit that a certain contractor will make on any one job, in
thousands of e, is a random variable X with probability density function given by
f (x) =

 1
 04
(4x − x3 )
for 0 < x < 2
elsewhere
a) Find the expectation value and the standard deviation of X
b) On average how much profit can this contractor be expected to make per job?
c) Should you be surprised if he makes a profit of less than e600 on his next
job? Why?
d) Should you be surprised if the contractor makes a profit of less than e600 on
average over the next 2, 000 jobs ? Why ?
4. Suppose that the time interval (in minutes) between cars arriving at certain toll
booth is an exponential random variable with expectation value 2 (minutes). Find
the probability that
a) there will be a wait of at least 3 minutes between the first and the second car
b) there will be a wait of at least 3 minutes between each of the first 6 cars
arriving that day
5. Show that the exponential random variable can be used to describe memoryless
processes, ie for t, s ≥ 0 one has
P (X > t + s|X > s) = P (X > t)
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Lecturer: Stefan Sint, sint@maths.tcd.ie
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