Math Help - Finding a probability function R (the remainder when divided by a distribution)

1. Finding a probability function R (the remainder when divided by a distribution)

Hi everyone, I am really unsure of how to do this question.

Let X have a geometric distribution with f(x) = p(1-p)^x for x = 0,1,2,....
Find the probability function of R, the remainder when X is divided by 4.

I don't know where to start, but I'm suspecting that you would construct a geometric series with r, and find an expression for f(r)... Not really sure.

2. Re: Finding a probability function R (the remainder when divided by a distribution)

If $0\le r < 4$, then $P(R = r) = P(X = r) + P(X = 4 + r) + P(X = 8 + r) + \dots =$ $p(1 - p)^r + p(1 - p)^{4+r} + p(1 - p)^{8+r} + \dots$, which is a geometric series.

3. Re: Finding a probability function R (the remainder when divided by a distribution)

So taking that geometric series, I can find the sum of the geometric series,which will express P (R = r)?

That is, we have P (R = r) = p(1-p)^r/ 1-((1-p)^4+r)) ?

THanks!

4. Re: Finding a probability function R (the remainder when divided by a distribution)

Originally Posted by KelvinScale
That is, we have P (R = r) = p(1-p)^r/ 1-((1-p)^4+r)) ?
There is no +r in the exponent: $P(R = r) =\frac{p(1-p)^r}{1-(1-p)^4}$.