An e-mail message will arrive at a time uniformly...

An e-mail message will arrive at a time uniformly distributed between 9: 00 A.M. and 11: 00 A.M. You check e-mail at 9: 15 A.M. and every 30 minutes afterward. (a) What is the standard deviation of arrival time (in minutes)? (b) What is the probability that the message arrives less than 10 minutes before you view it? (c) What is the probability that the message arrives more than 15 minutes before you view it?

An e-mail message will arrive at a time uniformly distributed between 9: 00 A.M. and 11: 00 A.M. You check e-mail at 9: 15 A.M. and every 30 minutes afterward.
(a) What is the standard deviation of arrival time (in minutes)?
(b) What is the probability that the message arrives less than 10 minutes before you view it?
(c) What is the probability that the message arrives more than 15 minutes before you view it?

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Key Concepts

Uniform Distribution

A uniform distribution is a type of probability distribution in which all outcomes are equally likely within a given continuous interval. Its probability density function is constant over the interval, which simplifies the process of calculating probabilities, means, and variances for any subinterval within the domain.

Standard Deviation of a Uniform Distribution

The standard deviation of a continuous uniform distribution is derived from its variance formula, where the variance is given by (b - a)²/12 for an interval from a to b. The standard deviation is simply the square root of this variance, and it quantifies the average deviation of the random variable from its mean.

Probability Calculations for Continuous Random Variables

Calculating probabilities for continuous random variables, such as those following a uniform distribution, involves integrating or calculating the proportion of the interval where the event of interest occurs relative to the total interval length. This technique is essential when determining the likelihood of events occurring within specific subintervals.

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