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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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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 ) = 1 3 e t / 3 ,        [ 0 , ]

(a) P ( t < 2 )

(b) P ( t 2 )

(c) P ( 1 < t < 4 )

(d) P ( t > 3 )

(a)

To determine

To calculate: The probability P(t<2), for the probability density function f(t)=13et/3 over the interval [0,)

Explanation

Given Information:

The function, f(t)=13et/3 is a probability density function over the interval [0,).

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)=13et/3

The function is a probability density function over the interval [0,).

Therefore, f(x) is continuous and non-negative over the interval [0,) and is graphed as shown below:

Now, the probability that t is less than 2 implies that t lies between 0 and 2 is computed as,

P(t<2)=02f(t)dt=0213et/3dt

(b)

To determine

To calculate:

The probability P(t2), for the probability density function f(t)=13et/3 over the interval [0,).

(c)

To determine

To calculate: The probability P(1<t<4), for the probability density function f(t)=13et/3 over the interval [0,).

(d)

To determine

To calculate: The probability P(t>3), for the probability density function f(t)=13et/3 over the interval [0,).

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