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Chapter 4: Problem 10

The probability density function of the length of a cutting blade is \(f(x)=1.25\) for \(74.675.2)\) (c) If the specifications for this process are from 74.7 to 75.3

Short Answer

Expert verified
(a) 0.25; (b) 0.5; (c) Probability within specifications is 0.75.

Step by step solution

01

Understanding the PDF

The given probability density function (pdf) is constant, i.e., 1.25, for the interval 74.6 to 75.4 millimeters. Since it is a uniform distribution over this interval, the area under the curve over a subinterval corresponds to the probability of that subinterval.
02

Find P(X

The probability that a continuous random variable takes a value less than a certain value is the area under the pdf curve to the left of that value. For this problem, calculate the area from 74.6 to 74.8: \[P(X < 74.8) = ext{length of interval} \times ext{height of pdf} = (74.8 - 74.6) \times 1.25 = 0.2 \times 1.25 = 0.25\]
03

Find P(X75.2)

This step involves two disjoint intervals. Compute the probability for each interval separately, then sum them. 1. For \(X < 74.8\): already calculated as 0.25.2. For \(X > 75.2\), calculate the area from 75.2 to 75.4:\[P(X > 75.2) = (75.4 - 75.2) \times 1.25 = 0.2 \times 1.25 = 0.25\]The total probability is the sum of both: \[P(X<74.8 \, \text{or} \, X>75.2) = P(X<74.8) + P(X>75.2) = 0.25 + 0.25 = 0.5\]
04

Check process specifications

Evaluate if the full range is within the given specification limits from 74.7 to 75.3. 1. The probability period from 74.7 to 75.3 must be calculated:\[P(74.7 < X < 75.3) = (75.3 - 74.7) \times 1.25 = 0.6 \times 1.25 = 0.75\] This reflects the proportion of the distribution that meets the specifications.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Uniform Distribution
In probability theory, a uniform distribution is one in which all outcomes are equally likely. Specifically, for a continuous uniform distribution, this means that every value within a certain interval has the same probability of occurring. For our exercise, we examine a uniform distribution that extends from 74.6 to 75.4 millimeters. Within this range, every length measurement of the cutting blade is equally probable, with a probability density function (pdf) set at a constant value of 1.25. This constant density allows easy calculation of probabilities by simply multiplying the interval length by the height of the pdf.
Continuous Random Variable
A continuous random variable is one that can assume an infinite number of values within a given range. Unlike discrete random variables, which take on specific values, continuous variables can represent any value within an interval. In this exercise, the length of a cutting blade is a continuous random variable. It can take any value between 74.6 and 75.4 millimeters. The probability of the variable taking an exact value is theoretically zero because of the infinite possibilities. Hence, we focus on the probability of the variable falling within certain intervals using the probability density function.
Probability Calculation
Probability calculations for a continuous random variable involve finding the area under the probability density function (pdf) curve within the desired interval. Here, the probability that the length of the cutting blade is less than 74.8 millimeters is calculated as follows:
  • First, determine the interval from which the probability is calculated; in this case, from 74.6 to 74.8.
  • Next, multiply the interval length by the height of the pdf (1.25) to get the area under the curve, yielding a probability of 0.25 for this part.
  • Similarly, for other calculations, add the probabilities of disjoint intervals to find cumulative probabilities, as seen in the calculation of \(P(X<74.8 \text{ or } X>75.2)\).
Remember, this method simplifies the calculation process in a uniform distribution due to the constant pdf value.
Process Specifications
Process specifications refer to the desired limits within which a process should operate to meet quality standards. In our context, the cutting blade's length should ideally be within 74.7 to 75.3 millimeters. The probability that a randomly selected blade length is within these limits can be calculated to ensure quality control. This involves finding the density function value for the interval 74.7 to 75.3:
  • Calculate the length of this interval, 0.6 millimeters.
  • Multiply by the pdf's constant value (1.25) to find the probability of 0.75.
This probability reflects the proportion of blades meeting the specifications, which is crucial for maintaining the desired quality levels in manufacturing.

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