Key Terms

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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 = $\frac{1}{μ}$ , 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 = $\frac{1}{m}$ and the standard deviation is σ = $\frac{1}{m}$ . The probability density function is f(x) = me^−mx, x ≥ 0, and the cumulative distribution function is P(X ≤ x) = 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 $P (X = k) = \frac{λ^{k} e^{- λ}}{k!}$ .

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 = $\frac{a + b}{2}$ and the standard deviation is $σ = \sqrt{\frac{{(b - a)}^{2}}{12}}$ . The probability density function is f(x) = $\frac{1}{b - a}$ for a < x < b or a ≤ x ≤ b. The cumulative distribution is P(X ≤ x) = $\frac{x - a}{b - a}$ .

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