Sections

Key Terms
# 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*= $\frac{1}{\mu}$ , 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 probabilitiesThis 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*k*events occurring in one unit time is equal to $P(X=k)=\frac{{\lambda}^{k}{e}^{-\lambda}}{k!}$.

- 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 \sigma =\sqrt{\frac{{\left(b-a\right)}^{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}.