Question

Exercise 31.1. Rope checks. In a certain manufacturing process, an automated quality control computer checks 10 yards of rope at a time. If no defects are detected in that 10-yard section, that portion of the rope is passed on. However, if there is a defect detected, a person will have to check the rope over more carefully to determine where (measured from the left side in yards) the defect is. If exactly 1 defect is detected in a rope section, we would like to find the probabilities for its location.

a. Why is this a Continuous Uniform problem?

b. What does X represent in this scenario?

c. What are the parameters in this scenario?

d. What is the expected value for the location of the defect?

e. What is the standard deviation?

f. What is the probability density function? Write it in function notation.

g. What is the cumulative distribution function? Write it in function notation.

h. Find P(X > 8). i. Find P(2.3 ≤ X ≤ 5.2) j. Find P(X < 2|X < 5)

i. Find the probability that a defect is within 1 standard deviation of the expected value. (Label this k.)

Answer 1

Answer:-

Given that:-

Exercise 31.1. Rope checks. In a certain manufacturing process, an automated quality control computer checks 10 yards of rope at a time. If no defects are detected in that 10-yard section, that portion of the rope is passed on. However, if there is a defect detected, a person will have to check the rope over more carefully to determine where (measured from the left side in yards) the defect is. If exactly 1 defect is detected in a rope section, we would like to find the probabilities for its location.Here X is the location at which defect is detected in a rope section of 10 yards

a. Why is this a Continuous Uniform problem?

Here

we know that it defects is detected in a rope section of 10 yards ,then location of defect is any where in a rope section of 10 yards All given 10 yards are have same chance is found defect .

b. What does X represent in this scenario?

X: Here X is the location at which defect is detected in a rope section of 10 yards

c. What are the parameters in this scenario?

Here i.e., a = 0 & b = 10

d. What is the expected value for the location of the defect?

Expected value of the location of the defect

e. What is the standard deviation?

standard Deviation -(SD):-

f. What is the probability density function? Write it in function notation.?

Probability density function:-

g. What is the cumulative distribution function? Write it in function notation.?

Cumlative Distribution function:-

h. Find P(X > 8). i. Find P(2.3 ≤ X ≤ 5.2) j. Find P(X < 2|X < 5)?

i. Find the probability that a defect is within 1 standard deviation of the expected value. (Label this k.)?

i.

   (using CDF function)

  

j.

   (Because using CDF function)