2. (10 points) Suppose you have a probability distribution P, supported on the positive integers 0,...,n,...

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2. (10 points) Suppose you have a probability distribution P, supported on the positive integers 0,...,n, and a second probability distribution Q supported on 0,...,m. Consider P as being represented by a vector indexed by 0,...,n, and Q as being a vector indexed by 0,...,m. Show that if we draw a sample p+P and independently draw a sample q+Q, then the sum p+q will be a positive integer in the range 0,...,n+m that is distributed according to the convolution of the distributions P*Q (when distributions are represented as vectors). 2. (10 points) Suppose you have a probability distribution P, supported on the positive integers 0,...,n, and a second probability distribution Q supported on 0,...,m. Consider P as being represented by a vector indexed by 0,...,n, and Q as being a vector indexed by 0,...,m. Show that if we draw a sample p+P and independently draw a sample q+Q, then the sum p+q will be a positive integer in the range 0,...,n+m that is distributed according to the convolution of the distributions P*Q (when distributions are represented as vectors).