Waiting Time The waiting time t for service in a store is...

Waiting Time The waiting time $t$ for service in a store is exponentially distributed with a mean of 5 minutes.
(a) Find the probability density function of the random variable $t$
(b) Find the probability that $t$ is within one standard deviation of the mean.

Key Concepts

Cumulative Distribution Function (CDF)

The cumulative distribution function gives the probability that a random variable is less than or equal to a particular value. For the exponential distribution, the CDF is derived from the PDF and is used to compute probabilities over intervals of the variable.

Mean and Standard Deviation Relationship

In the exponential distribution, the mean is the reciprocal of the rate parameter, and uniquely, the standard deviation is equal to the mean. This property simplifies calculations of spread and probability over symmetric intervals centered around the mean.

Exponential Distribution

The exponential distribution is a continuous probability distribution often used to model waiting times or the time until an event occurs. It is memoryless, meaning that the probability of an event occurring in the future is independent of how much time has already elapsed.

Probability Density Function (PDF)

The probability density function specifies the likelihood of a random variable taking on a specific value. For the exponential distribution, the PDF is defined using a rate parameter, which is the reciprocal of the mean. This function is fundamental to determining probabilities for continuous random variables.

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