Exponential Distributions: Parameters, Probabilities, and Memorylessness

All flashcards

How are the mean and decay parameter related for $X\sim\operatorname{Exp}(m)$?

The mean is $\mu=1/m$, so the decay parameter is $m=1/\mu$.

How does the standard deviation relate to the mean in an exponential distribution?

They are equal: $\sigma=\mu=1/m$.

What does the exponential probability density function represent?

The function $f(x)=me^{-mx}$ gives density height; probabilities are areas under the curve.

What is the memoryless property of an exponential random variable?

For $r,t\geq0$, $P(X>r+t\mid X>r)=P(X>t)$; elapsed waiting time does not change the remaining-wait distribution.

How are exponential waiting times related to Poisson event counts?

Under a constant-rate, independent-event model, interarrival times are exponential and counts over intervals are Poisson.

Quiz questions

1. For $X\sim\operatorname{Exp}(m)$ and $x\geq0$, which expression gives $P(X\leq x)$?Answer: $1-e^{-mx}$
2. For $X\sim\operatorname{Exp}(m)$ and $0\leq a<b$, which expression gives $P(a<X<b)$?Answer: $e^{-ma}-e^{-mb}$
3. For an exponential waiting time $X$, which equation expresses memorylessness for $r,t\geq0$?Answer: $P(X>r+t\mid X>r)=P(X>t)$
4. Under a constant-rate model with independent events, what distribution describes the number of events in a fixed interval?Answer: A Poisson distribution, with mean equal to the event rate times the interval length.
5. For $X\sim\operatorname{Exp}(m)$, what value $k$ has cumulative probability $P(X\leq k)=A$, where $0<A<1$?Answer: $k=\frac{-\ln(1-A)}{m}$