Key Terms

conditional probability
the likelihood that an event will occur given that another event has already occurred
decay parameter
the description of the rate at which probabilities decay to zero for increasing values of x

It is the value m in the probability density function f(x) = me(–mx) of an exponential random variable. It is also equal to m = 1μ1μ , where μ is the mean of the random variable.
exponential distribution
a continuous random variable (RV) that appears when we are interested in the intervals of time between some random events, for example, the length of time between emergency arrivals at a hospital; the notation is X ~ Exp(m)

The mean is μ = 1m1m and the standard deviation is σ = 1m1m. The probability density function is f(x) = me−mx, x ≥ 0, and the cumulative distribution function is P(Xx) = 1 − e−mx.
memoryless property
or an exponential random variable X, the statement that knowledge of what has occurred in the past has no effect on future probabilities

This means that the probability that X exceeds x + k, given that it has exceeded x, is the same as the probability that X would exceed k if we had no knowledge about it. In symbols we say that P(X > x + k|X > x) = P(X > k).
Poisson distribution
a distribution function that gives the probability of a number of events occurring in a fixed interval of time or space if these events happen
with a known average rate and independently of the time since the last event; if there is a known average of λ events occurring per unit time, and these events are independent of each other, then the number of events X occurring in one unit of time has the Poisson distribution
The probability of k events occurring in one unit time is equal to P(X=k)=λkeλk!P(X=k)=λkeλk!.
uniform distribution
a continuous random variable (RV) that has equally likely outcomes over the domain, a < x < b. Notation—X ~ U(a,b)

The mean is μ = a+b2a+b2 and the standard deviation is σ=(ba)212σ=(ba)212. The probability density function is f(x) = 1ba1ba for a < x < b or axb. The cumulative distribution is P(Xx) = xabaxaba.