Discrete Convolutions

[bookworm121]

How would you go about convolving two discrete distributions that look something like this:

Number	Probability
0	0.1
1	0.2
2	0.7

Number	Probability
0	0.1
1	0.2
2	0.3
3	0.3
4	0.1

 

[DrPhil]

How would you go about convolving two discrete distributions that look something like this:

Number	Probability
0	0.1
1	0.2
2	0.7

Number	Probability
0	0.1
1	0.2
2	0.3
3	0.3
4	0.1

How far have you gotten? We need to see your work to know where you need help.

I will assume you need help setting it up. You should draw the two distributions. Flip one and mentally shift it over the other, taking the sum of products where the distributions overlap.

The mathematical expression for that is

\(\displaystyle \displaystyle (f*g)_n = \sum_m \left(f(m) g(n-m)\right) \)

 

[HallsofIvy]

Since the first distribution is only defined for n= 0, 1, and 2, while the second is defined for n= 0, 1, 2, 3, and 4, extend the first to be 0 at both 3 and 4.
Write them as
\(\displaystyle \begin{array}{ccccc}0.1 & 0.3 & 0.3 & 0.2 & 0.1 \\ 0 & 0 & .7 & .2 & .1 \end{array}\)
Multiply the two numbers in each column and add.