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MA2B, HOMEWORK 3 SOLUTIONS
Problem 1: 3.1.10
(a) The random variable
S
n
is binomially distributed with distribution Bin(n,p)
because it is equal to the number of successes in n independent Bernoulli (p) trials.
(b)The random variable
T
m
is binomially distributed with distribution Bin(m,p)
because it is equal to the number of successes in m independent Bernoulli (p) trials.
(c) The random variable
S
n
+
T
m
is binomially distributed with distribution Bin(n+m,p)
because it is equal to the number of successes in n+m independent Bernoulli (p)
trials.
(d) Let
X
i
be the indicator function that there is a success on the i
th
trial. We
know from the statement of the problem that the outcome of any particular trial
is independent of the results of all the other trials, so that the random variables
X
1
,...,X
n
+
m
are independent. Now we can use two results about independent ran
dom variables that may be found in Pitman’s book on page 154. The Frst is that
disjoint blocks of random variables are independent. Thus we can deFne two joint
random variables Y and Z as follows:
Y
=(
X
1
,...,X
n
)
Z
=(
X
n
+1
,...,X
n
+
m
)
and we are guaranteed that Y and Z are independent. Note that Y and Z are
random variables whose values are binary vectors of length n and m respectively.
Now we can use the other result from page 154 that functions of independent
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 Winter '08
 Makarov,N
 Differential Equations, Statistics, Equations, Bernoulli, Binomial, Probability

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