Question

The number of phone calls coming into a telephone switchboard during each minute was recorded during an entire hour. During 30 of the 1-minute intervals there were no calls, during 20 intervals there was one call, and during 10 intervals there were two calls. A 1-minute interval is to be selected at random and the number of calls noted. Let $X$ be the outcome. Then, $X$ is a random variable taking on the values $0,1,$ and $2.$ (a) Write out a probability table for $X.$ (b) Compute $E(X).$ (c) Interpret $E(X).$

Name: Problem 6, Chapter 12: Calculus and Its Applications
Uploaded: 2021-07-01T21:09:28-08:00
Duration: 2 min
Channel: Kaitlin Yaeger

   The number of phone calls coming into a telephone switchboard during each minute was recorded during an entire hour. During 30 of the 1-minute intervals there were no calls, during 20 intervals there was one call, and during 10 intervals there were two calls. A 1-minute interval is to be selected at random and the number of calls noted. Let $X$ be the outcome. Then, $X$ is a random variable taking on the values $0,1,$ and $2.$
(a) Write out a probability table for $X.$
(b) Compute $E(X).$
(c) Interpret $E(X).$

Calculus and Its Applications

Larry J. Goldstein,… 14th Edition

Chapter 12, Problem 6

Instant Answer

Solved by Expert Kaitlin Yaeger

07/01/2021

Step 1

The random variable $X$ can take on the values $0, 1,$ and $2$. The probabilities for these values are given by the number of 1-minute intervals with that number of calls divided by the total number of 1-minute intervals in an hour (which is 60). So, the Show more…

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The number of phone calls coming into a telephone switchboard during each minute was recorded during an entire hour. During 30 of the 1-minute intervals there were no calls, during 20 intervals there was one call, and during 10 intervals there were two calls. A 1-minute interval is to be selected at random and the number of calls noted. Let $X$ be the outcome. Then, $X$ is a random variable taking on the values $0,1,$ and $2.$ (a) Write out a probability table for $X.$ (b) Compute $E(X).$ (c) Interpret $E(X).$

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Key Concepts

Random Variable

A random variable is a numerical outcome of a random process. It assigns a number to each outcome in a sample space, allowing for quantitative analysis of random phenomena.

Probability Distribution

A probability distribution details the likelihood of each possible value that a random variable can take. It is typically represented in a table or formula, showing the probabilities corresponding to each outcome.

Expected Value

The expected value is a measure of the central tendency of a random variable, calculated as the sum of each possible outcome multiplied by its probability. It represents the long-run average if the random process were repeated many times.

Interpretation of Expected Value

The interpretation of expected value is that it gives an overall summary or 'average' outcome of a random process. It indicates what value one can generally anticipate over a large number of trials, even though individual outcomes may vary.

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