ERG2040A Probability Models and Applications
Example Sheet 3
1. (The Geometric Distribution) Find the mode, median, mean and variance of
the geometrically distributed random variable T , viz. P (T = k) = q k p; 8k 2 N
where p + q = 1 and p; q ? 0. Show that the geometric distribution is memoryless,
i.e. P (T = k + l j T ? l) = P (T = k).
2. (The Exponential Density) The exponential density function, viz.
f(x) =
ae
–e
median, mean and variance from first principle.
(b) Find its generating function, and thereby its skewness and kurtosis.
(c) Show that the exponential density is memoryless. Can you express the mem
oryless property as an independence relation between appropriately defined
random variables?
3. (Binomial and Poisson Distributions) It is instructive to consider a binomial
distributed random variable S ¸ b(n; p; k) as the number of successes out of n
Bernoulli trials with success rate p.
(a) Find the mode, median, mean and variance of S.
(b) Point events (e.g. births, deaths, arrivals) occur randomly in the interval
(
By treating the resulting Poisson points as successes among
a large number of Bernoulli trials conducted uniformly in the interval, show
that the point spacings are exponentially distributed. Derive also the resulting
Poisson distribution of X , the total number of points on the interval. Find its
mode, median, mean and variance.
(c) Learn Matlab and plot b(n; p; k), its Poisson approximation and the percentage
error for the following sets of (n; p) values: (i) (100,.1); (ii) (100,.01); (iii)
(1000,.1).
4. (Justifying the Poisson Assumption) The number of fouled toilet incidents in
the HSH Engineering Building is found to follow a Poisson distribution with a mean
rate of once a day.
(a) What is the chance that more than two foul toilets are found on a particular
day?
(b) Which do you think explains the Poisson property better: a small number of
frequent culprits or a large number of occasional absentminded users?
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5. (Scalar Quantization) A random analogue signal is sampled and uniformly
quantized to N bits. Assuming the analogue samples are uniformly distributed over
the dynamic range, show that the average error is 0 while the root mean squared
(RMS) error is inversely proportional to 2 N . Convince yourself that as N increases,
the uniformity assumption becomes less essential for the results to hold.
6. (Variance) X and Y are two random variables with means ¯X and ¯ Y , and vari
ances oe 2
X and oe 2
Y respectively.
(a) Derive the variances of cX , cXY and cX + dY where c; d are constants and
X; Y are independent.
(b) Construct an example for which the variance of X + Y is smaller than both
oe 2
X and oe 2
Y .
7. (N draws with replacement from a set of N) A generous professor teaches
a class of N and gives a random student a red packet in each lecture. What is
the expected number of miserable students (without a single red packet) after N
lectures? (Practise the indicator variable method.)
8. (Card Collection) How many cards on average need be collected for a full set
of N different designs available with equal probabilities?
9. (Summing A Random Number of Random Variables) What is the expectation
of S = X 1 + X 2 + : : : + XN where the X i 's are uniformly distributed in [0; 1] and
N is the the outcome of a fair die roll?
10. (Income Distribution and Inequality) Assume the proportion of the working
population in Hong Kong with income more than HK$x is well approximated by
the Gamma distribution
G(x; –; n) =
ae
e Ÿ x ! 1
0 otherwise
where –; n are positive constants.
A recent survey shows that while the average income has remained constant at
HK$12k per month, the modal income has dropped from HK$9k in 1997 to HK$8k
in 1999 due to economic setback.
(a) Plot the two income distributions using Matlab.
(b) Assuming the working population is three millions, how many more people are
now below the HK$5,000 per month mark?
(c) Estimate the change in the median income.
(d) Estimate the values of the Gini index before and after the setback.
WaiYin Ng
Febrary 2000
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(
geocities.com/hk/ugly_hui/csc) (
geocities.com/hk/ugly_hui) (
geocities.com/hk)