Length of a Telephone Call The length of a telephone call...

Length of a Telephone Call The length of a telephone call (in minutes), $t$ , for a certain town is a continuous random variable with probability density function defined by $f(t)=3 t^{-4},$ for $t$ in $[1, \infty)$. Find the probabilities for the following situations. (a) The call lasts between 1 and 2 minutes. (b) The call lasts between 3 and 5 minutes. (c) The call lasts longer than 3 minutes.

Length of a Telephone Call The length of a telephone call (in
minutes), $t$ , for a certain town is a continuous random variable
with probability density function defined by
$f(t)=3 t^{-4},$ for $t$ in $[1, \infty)$.
Find the probabilities for the following situations.
(a) The call lasts between 1 and 2 minutes.
(b) The call lasts between 3 and 5 minutes.
(c) The call lasts longer than 3 minutes.

Key Concepts

Integration to Find Probabilities

For continuous random variables, probabilities that the variable falls within a specific interval are determined by integrating the pdf over that interval. This process accumulates the relative likelihoods over the range of interest, yielding the probability of the event.

Improper Integral

An improper integral arises when the range of integration is infinite or when the function has an infinite discontinuity. In the context of probability, improper integrals are used to compute probabilities over unbounded intervals, ensuring that the total probability over the entire range sums to 1.

Continuous Random Variable

A continuous random variable can assume any value within a continuum or interval. Probabilities are assigned to intervals rather than specific outcomes, and its behavior is characterized by a probability density function rather than a probability mass function, which is used for discrete random variables.

Probability Density Function (pdf)

A probability density function (pdf) defines the relative likelihood that a continuous random variable takes on a given value. It must be non-negative for all values and must integrate to 1 over the entire range of possible values. The pdf is used to calculate the probability over any interval by integration.

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