08Ma2bSol3 - MA2B HOMEWORK 3 SOLUTIONS Problem 1 3.1.10(a...

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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 first is that disjoint blocks of random variables are independent. Thus we can define 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.
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