   # Waiting Time The waiting time t (in minutes) for patients arriving at a health clinic is described by the probability density function f ( t ) = 1 12 e − t / 12 , [ 0 , ∞ ) Find the probability that a patient will wait (a) no more than 5 minutes and (b) between 9 and 12 minutes. ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
Publisher: Cengage Learning
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
Publisher: Cengage Learning
ISBN: 9781305860919
Chapter 9, Problem 37RE
Textbook Problem
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## Waiting Time The waiting time t (in minutes) for patients arriving at a health clinic is described by the probability density function f ( t ) = 1 12 e − t / 12 , [ 0 , ∞ ) Find the probability that a patient will wait (a) no more than 5 minutes and (b) between 9 and 12 minutes.

(a)

To determine

To calculate: The probability that the patient will not wait more than 5 minutes at the health clinic for the waiting time defined by the probability density function f(t)=112et/12 over the interval [0,).

### Explanation of Solution

Given Information:

The probability density function of the waiting time for the patients who arrive at a health clinic is defined as,

f(t)=112et/12 over the interval [0,).

Here, t is the waiting time in minutes.

Formula used:

If f is function of a continuous random variable x, then the probability that x lies in the interval [c,d] is,

P(cxd)=cdf(x) dx

Integration formula,

eaxdx=eaxa+C

Where, C is a constant and a is a real number.

The Fundamental theorem,

abf(x) dx=F(b)F(a)

Calculation:

Consider the provided probability density function of the waiting time for the patients who arrive at a health clinic defined as,

f(t)=112et/12 over the interval [0,).

The required probability that the patient will not wait more than 5 minutes is P(0t5).

Use the formula P(cxd)=cdf(x) dx to calculate the required probability P(0t5)

(b)

To determine

To calculate: The probability that the patient waits between 9 and 12 minutes at the health clinic for the waiting time defined by the probability density function f(t)=112et/12 over the interval [0,).

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