   Chapter 9.2, Problem 26E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

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

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Finding a Probability In Exercises 19-26, sketch the graph of the probability density function over the indicated interval and find each probability. See Example 3. f ( t ) = 3 256 ( 16 − t 2 ) ,        [ − 4 , 4 ] (a) P ( t < − 2 ) (b) P ( t > 2 ) (c) P ( − 1 < t < 1 ) (d) P ( t > − 2 )

(a)

To determine

To calculate: The probability P(t<2), for the probability density function f(t)=3256(16t2) over the interval [4,4].

Explanation

Given Information:

The function f(t)=3256(16t2) is a probability density function over the interval [4,4].

Formula used:

Consider a function f of a continuous random variables x whose set of values is the interval [a,b], A function is a probability density function when it is non-negative and continuous on the interval [a,b] and when abf(x)dx=1 and the probability P(a<x<b)=abf(x)dx.

Calculation:

Consider the function,

f(t)=3256(16t2)

The function is a probability density function over the interval [4,4].

Therefore, f(x) is continuous and non-negative over the interval [4,4] and is graphed as shown below:

Now, the probability that t is less than 2 implies that t lies between 4 and 2 is computed as

P(t<2)=42f(t)dt=423256(16t2)dt=325642(16t2)dt=3256{

(b)

To determine

To calculate: The probability P(t>2), for the probability density function f(t)=3256(16t2) over the interval [4,4]

(c)

To determine

To calculate: The probability P(1<t<1), for the probability density function f(t)=3256(16t2) over the interval [4,4]

(d)

To determine

To calculate: The probability P(t>2), for the probability density function f(t)=3256(16t2) over the interval [4,4].

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