You are second in line at the bank. (Does anyone go to a bank anymore?) Assume that the time a teller takes to deal with a customer is exponentially distributed with mean 5 minutes.
You’ve been waiting an absurd fifteen minutes already. How long should you expect to wait after this to be seen?
If you’ve waited fifteen minutes, how long do you expect to take, in total, at the bank?
There are two water fountains in front of you at the gym, but they’re both occupied. Assume the time it takes a gym-goer to satiate their thirst is exponentially distributed with mean 20 seconds.
How long should you expect to wait until a water fountain is available?
How long should you expect to be at the water fountain total?
What is the probability that you are the last person at the water fountain? (This one is tricky!)
Let \(X \sim N(0, 1/2)\). Write \(P(X \geq 1/2)\) in terms of \(\Phi(z)\), the cdf of the standard normal.
You’re trying to send your friend an \(n\)-bit message (meaning a sequence of length \(n\) consisting of only 0’s and 1’s) over a ‘noisy’ communication channel. If bit \(i\) is \(x_i\), then your friend receives \(x_i + \epsilon_i\), where \(\epsilon_i\) is an \(N(0, 1/2)\) variable.
Your friend will interpret anything over \(1/2\) as a \(1\), and anything under \(1/2\) as a \(0\). What is the probability that your first bit is correctly received?
What is the probability that all \(n\) bits are correctly received, assuming that the noises are independent?