The length of time of calls made to a support helpline...

The length of time of calls made to a support helpline follows an exponential distribution with an average duration of 40 minutes so that $\lambda=1 / 40=.025 .$ a. What is the probability that a call to the helpline lasts less than 5 minutes? b. What is the probability that a call to the helpline lasts more than 50 minutes? c. What is the probability that a call lasts between 20 and 30 minutes? d. Tchebysheff's Theorem says that the interval $40 \pm 2(40)$ should contain at least $75 \%$ of the population. What is the actual probability that the call times lie in this interval?

The length of time of calls made to a support helpline follows an exponential distribution with an average duration of 40 minutes so that $\lambda=1 / 40=.025 .$
a. What is the probability that a call to the helpline lasts less than 5 minutes?
b. What is the probability that a call to the helpline lasts more than 50 minutes?
c. What is the probability that a call lasts between 20 and 30 minutes?
d. Tchebysheff's Theorem says that the interval $40 \pm 2(40)$ should contain at least $75 \%$ of the population. What is the actual probability that the call times lie in this interval?

Key Concepts

Tchebysheff's Theorem

Tchebysheff's Theorem provides a way to estimate the proportion of observations lying within a certain number of standard deviations from the mean for any distribution, regardless of its shape. It sets a minimum bound on the probability of observations falling within a specified interval around the mean, making it a useful tool for understanding dispersion in a distribution without needing to assume a particular form.

Interval Probability Calculation

Calculating the probability that a random variable falls within an interval in a continuous distribution involves taking the difference between the CDF values at the endpoints of the interval. For the exponential distribution, this involves computing the probability from the lower bound and subtracting it from the probability up to the upper bound.

Survival Function

The survival function, which is the complement of the CDF, represents the probability that the event has not occurred by a certain time, computed as S(x) = e^(-?x) for the exponential distribution. It is particularly useful when calculating the probability that the duration exceeds a certain threshold.

Exponential Distribution

The exponential distribution is a continuous probability distribution that describes the time between events in a Poisson process. It is often used to model lifetimes or waiting times, such as the duration of calls, and is characterized by a constant hazard rate, meaning that the probability of an event occurring in the next interval is independent of how much time has already elapsed.

Rate Parameter (?)

The rate parameter, denoted as ?, is a key component of the exponential distribution and represents the average number of events per unit time. It is the reciprocal of the mean, and it greatly influences the shape of the distribution; low values of ? indicate a longer average waiting time, while higher values signify more frequent events.

Cumulative Distribution Function (CDF)

The Cumulative Distribution Function (CDF) for the exponential distribution gives the probability that the variable takes on a value less than or equal to a specific point. It is calculated as F(x) = 1 - e^(-?x), which is used to determine probabilities for intervals that start from zero up to a specific time.

Transcript

Please give Ace some feedback