Final answer:
The population's mean for an exponential random variable with rate parameter λ is 1/λ. The exponential distribution is used to model the time between random events and is related to the Poisson distribution, where the latter counts events per unit time.
Step-by-step explanation:
The population's mean for an exponential random variable with the rate parameter λ (lambda) is μ (mu), which is the reciprocal of λ. In mathematical terms, the population mean μ is equal to 1/λ. The exponential distribution is a continuous random variable (RV) that is often used to model the intervals of time between random events. For instance, it could represent the length of time between emergency arrivals at a hospital. The notation for an exponential distribution is X ~ Exp(m), where the mean μ is given by 1/m and m is the decay parameter equivalent to λ.
In the context of both the exponential and Poisson distributions, if the time between events follows an exponential distribution with mean μ, then the count of events per unit time follows a Poisson distribution with mean λ = 1/μ. This is because both distributions are mathematically related where the interval time between random events in the exponential distribution corresponds to the count of events in a fixed interval in the Poisson distribution.