A number x is selected at random from the interval [4,20]. The probability density function for x is given by the following function. Find the probability that a number selected is in the subinterval [5,17]. f(x)= for 4 ≤x≤20. 16' How is the probability that a number selected is in the subinterval [5,17] calculated? 1 ⒸA. Integrate 16 twice, then evaluate the integral over the limits 5 and 17. OB. Evaluate over the limits 5 and 17, then add. 16 OC. Integrate 16 then evaluate the integral over the limits 5 and 17. O D. Evaluate over the limits 5 and 17, then subtract. 16 The probability is
A number x is selected at random from the interval [4,20]. The probability density function for x is given by the following function. Find the probability that a number selected is in the subinterval [5,17]. f(x)= for 4 ≤x≤20. 16' How is the probability that a number selected is in the subinterval [5,17] calculated? 1 ⒸA. Integrate 16 twice, then evaluate the integral over the limits 5 and 17. OB. Evaluate over the limits 5 and 17, then add. 16 OC. Integrate 16 then evaluate the integral over the limits 5 and 17. O D. Evaluate over the limits 5 and 17, then subtract. 16 The probability is
Calculus: Early Transcendentals
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Chapter1: Functions And Models
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![### Probability Density Function and Interval Calculation
#### Problem Statement
A number \( x \) is selected at random from the interval \([4, 20]\). The probability density function for \( x \) is given by the following function. The task is to find the probability that a number selected is in the subinterval \([5, 17]\).
\[ f(x) = \frac{1}{16}, \quad \text{for } 4 \leq x \leq 20.\]
#### Question:
How is the probability that a number selected is in the subinterval \([5, 17]\) calculated?
#### Options:
A. Integrate
\[ \frac{1}{16} \]
twice, then evaluate the integral over the limits 5 and 17.
B. Evaluate
\[ \frac{1}{16} \]
over the limits 5 and 17, then add.
C. Integrate
\[ \frac{1}{16} \]
then evaluate the integral over the limits 5 and 17.
D. Evaluate
\[ \frac{1}{16} \]
over the limits 5 and 17, then subtract.
#### Solution:
Select the correct option to solve the problem. We need to find the probability that \( x \) is in a subinterval \([5, 17]\). This involves calculating the integral of the probability density function \( f(x) \) over this interval.
Since the probability density function is constant (\(\frac{1}{16}\)) over the whole interval from 4 to 20, we integrate \( \frac{1}{16} \) over the limits 5 and 17.
The correct option is **C. Integrate \(\frac{1}{16}\), then evaluate the integral over the limits 5 and 17.**
To solve this:
1. Integrate the constant function \( \frac{1}{16} \) over the interval \([5, 17]\):
\[ \int_{5}^{17} \frac{1}{16} \, dx = \left[ \frac{1}{16} x \right]_{5}^{17} \]
2. Evaluate the definite integral:
\[ \left[ \frac{1}{16} x \right]_{5}^{17} = \frac{1}{16} \left( 17 - 5 \right) =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc56b3bff-26b7-4f28-9b8d-5c65e8bb1b58%2F3104384f-3cfa-4a16-a7ba-e83fc0120088%2Fsmd3n6p_processed.png&w=3840&q=75)
Transcribed Image Text:### Probability Density Function and Interval Calculation
#### Problem Statement
A number \( x \) is selected at random from the interval \([4, 20]\). The probability density function for \( x \) is given by the following function. The task is to find the probability that a number selected is in the subinterval \([5, 17]\).
\[ f(x) = \frac{1}{16}, \quad \text{for } 4 \leq x \leq 20.\]
#### Question:
How is the probability that a number selected is in the subinterval \([5, 17]\) calculated?
#### Options:
A. Integrate
\[ \frac{1}{16} \]
twice, then evaluate the integral over the limits 5 and 17.
B. Evaluate
\[ \frac{1}{16} \]
over the limits 5 and 17, then add.
C. Integrate
\[ \frac{1}{16} \]
then evaluate the integral over the limits 5 and 17.
D. Evaluate
\[ \frac{1}{16} \]
over the limits 5 and 17, then subtract.
#### Solution:
Select the correct option to solve the problem. We need to find the probability that \( x \) is in a subinterval \([5, 17]\). This involves calculating the integral of the probability density function \( f(x) \) over this interval.
Since the probability density function is constant (\(\frac{1}{16}\)) over the whole interval from 4 to 20, we integrate \( \frac{1}{16} \) over the limits 5 and 17.
The correct option is **C. Integrate \(\frac{1}{16}\), then evaluate the integral over the limits 5 and 17.**
To solve this:
1. Integrate the constant function \( \frac{1}{16} \) over the interval \([5, 17]\):
\[ \int_{5}^{17} \frac{1}{16} \, dx = \left[ \frac{1}{16} x \right]_{5}^{17} \]
2. Evaluate the definite integral:
\[ \left[ \frac{1}{16} x \right]_{5}^{17} = \frac{1}{16} \left( 17 - 5 \right) =
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