   Chapter 8.5, Problem 13E

Chapter
Section
Textbook Problem

REM sleep is the phase of sleep when most active dreaming occurs. In a study, the amount of REM sleep during the first four hours of sleep was described by a random variable T with probability density function f ( t ) = { 1 1600 t if     0 ≤ t ≤ 40 1 20 − 1 1600 t if   40 < t ≤ 80 0 otherwise where t is measured in minutes. (a) What is the probability that the amount of REM sleep is between 30 and 60 minutes? (b) Find the mean amount of REM sleep.

(a)

To determine

To find: The probability that the amount of REM sleep is between 30 and 60 minutes.

Explanation

Given information:

Probability density function is f(t)={11,600t        if0t4012011,600tif40<t800                 otherwise .

Calculation:

Find the probability that the amount of REM sleep is between 30 and 60 minutes P(30T60) as shown below.

P(30T60)=3060f(t)dt=3040f(t)dt+4060f(t)dt=3040(t<

(b)

To determine

To find: The mean amount of REM sleep.

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