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.

Any insight would be helpful.

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

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

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!

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

Quote:

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: $\displaystyle P(R = r) =\frac{p(1-p)^r}{1-(1-p)^4}$.