Study for Final
Exponential distribution: P(a,b) = e^-lamba*a - e^-lambda*b. Which is same as integral of lambdae^-lambda*x from a to b.
Poission distribution: lambda*k e^-lambda / k! for k is integer. This for “what is probability that there are k arrivals given lambda and that it’s poisson”. You can get intervals from this by summing k’s. (so if you wanted probability that it was (8-10), you would plug in k=8,9,10 and add them.
Binomial distribution: (n choose x) p^x q^n-x. x must be integer. Sum across x’s to get probability of intervals.
For discrete distributions, there is typically no way to write the CDF with a formula. Just write each value down.
For continuous, just evaluate the integral of the density from zero to x. That should be your CDF.
Poission process any interval [t,t+s], number of arrivals is p(x) = (lambda*s)^x*e^-lambda*s/x!. (p173).
For any point t, waiting time from t to next arrival is lambda*e^-lambda*t.