The time (in minutes) between telephone calls at an...

The time (in minutes) between telephone calls at an insurance claims office has the following exponential probability distribution. \[f(x)=.50 e^{-.50 x} \quad \text { for } x \geq 0\] a. What is the mean time between telephone calls? b. What is the probability of having 30 seconds or less between telephone calls? c. What is the probability of having 1 minute or less between telephone calls? d. What is the probability of having 5 or more minutes without a telephone call?

The time (in minutes) between telephone calls at an insurance claims office has the following exponential probability distribution.
\[f(x)=.50 e^{-.50 x} \quad \text { for } x \geq 0\]
a. What is the mean time between telephone calls?
b. What is the probability of having 30 seconds or less between telephone calls?
c. What is the probability of having 1 minute or less between telephone calls?
d. What is the probability of having 5 or more minutes without a telephone call?

Key Concepts

Exponential Distribution

The exponential distribution is a continuous probability distribution that describes the time between events in a Poisson process, where events occur continuously and independently at a constant average rate. Its probability density function is typically given as f(x) = ?e^(–?x) for x ? 0, and it is widely used in scenarios such as reliability analysis, queuing theory, and survival analysis.

Mean (Expected Value) of the Exponential Distribution

The mean or expected value of the exponential distribution is the reciprocal of the rate parameter ?, given by E[X] = 1/?. This value represents the average waiting time between events, which is a key measure of central tendency in probability and statistics.

Cumulative Distribution Function (CDF)

The Cumulative Distribution Function (CDF) of the exponential distribution, F(x) = 1 – e^(–?x), gives the probability that the time between events is less than or equal to a certain value x. It is essential for calculating the likelihood of an event occurring within a specified time frame.

Survival Function

The survival function, defined as S(x) = e^(–?x), is the complement of the CDF and represents the probability that the waiting time until the next event exceeds a given value x. It is particularly useful in reliability studies and other applications where the focus is on the time until an event failure or occurrence.

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