# Variance and Expected value of internet connection

I am working on a probability/statistics problem!

The problem is as follows:

Your internet connection is very poor. It constantly alternates between being functional for x minutes and being down for y minutes. If you try to check your email at a random time, how long do you have to wait, on average, to get internet connection? What is variance of this waiting time?

I don't even know how I would approach this problem because the random time of x and y minutes do not have a probability that I would need to compute the expected value (which is summation of probability * value). I guess these are expected value of continuous variables, but then again I am very stuck.

Could someone guide me through this problem? thanks!

• I think $x$ and $y$ may be considered to be constants, and you arrive at a random time in the cycle. Then your waiting time $T$ is $0$ with probability $\frac{x}{x+y}$ and uniformly distributed on the interval $(0,y)$ with probability $\frac{y}{x+y}$. – André Nicolas Apr 18 '15 at 4:17

What is the probability that you try to check your email during the first $x$ minutes of the $(x+y)$-minute cycle? If you do that, then your waiting time is $0$.
What is the probability that you try to check your email when more than $w$ minutes, but fewer than $y$ minutes, remain of the $(x+y)$-minute cycle, where $0<w<y$? In that case, you need to wait more than $w$ minutes.
• Could the person who down-voted this explain what objections there are? ${}\qquad{}$ – Michael Hardy Apr 18 '15 at 17:27