   # Medicine The time t (in days) until recovery after a certain medical procedure is described by the probability density function f ( t ) = 1 4 t − 4 , [ 5 , 13 ] . Find the probability that a patient will take (a) no more than 6 days to recover and (b) at least 8 days to recover. ### 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 38RE
Textbook Problem
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## Medicine The time t (in days) until recovery after a certain medical procedure is described by the probability density function f ( t ) = 1 4 t − 4 , [ 5 , 13 ] .Find the probability that a patient will take (a) no more than 6 days to recover and (b) at least 8 days to recover.

(a)

To determine

To calculate: The probability that the patient will not take no more than 6 days to recover after a certain medical procedure where the time until recovery is defined by the probability density function f(t)=14t4 over the interval [5,13].

### Explanation of Solution

Given Information:

The probability density function of the time until recovery after a certain medical procedure is defined as,

f(t)=14t4 over the interval [5,13].

Here, t is the time until recovery in days.

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

The integration formula,

xndx=xn+1n+1+C

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

The Fundamental theorem,

abf(x) dx=F(b)F(a)

Calculation:

Consider the provided probability density function of the time until recovery after a certain medical procedure defined as,

f(t)=14t4 over the interval [5,13].

The required probability that the patient will take no more than 6 days to recover is P(5t6).

Use the formula P(cxd)=cdf(x) dx to calculate the required probability P(5t6)

(b)

To determine

To calculate: The probability that the patient will take at least 8 days to recover after a certain medical procedure where the time until recovery is defined by the probability density function f(t)=14t4 over the interval [5,13].

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