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The article "Response of \(\mathrm{SiC}_{\mathrm{I}} / \mathrm{Si}_{3} \mathbf{N}_{4}\) Composites Under Static and Cyclic Loading-An Experimental and Statistical Analysis" \((J\). of Engr. Materials and Technology, 1997: 186-193) suggests that tensile strength (MPa) of composites under specified conditions can be modeled by a Weibull distribution with \(\alpha=9\) and \(\beta=180\). a. Sketch a graph of the density function. b. What is the probability that the strength of a randomly selected specimen will exceed 175 ? Will be between 150 and 175 ? c. If two randomly selected specimens are chosen and their strengths are independent of one another, what is the probability that at least one has a strength between 150 and 175 ? d. What strength value separates the weakest \(10 \%\) of all specimens from the remaining \(90 \%\) ?

Step by step solution

01

Understanding the Weibull Distribution

The Weibull distribution is given by the probability density function (pdf): \[ f(x; \alpha, \beta) = \frac{\alpha}{\beta} \left(\frac{x}{\beta}\right)^{\alpha - 1} e^{-(x/\beta)^\alpha} \textrm{ for } x \geq 0,\]where \(\alpha\) is the shape parameter and \(\beta\) is the scale parameter. For this problem, \(\alpha = 9\) and \(\beta = 180\).

02

Sketching the Density Function

Using the Weibull pdf provided, plot the graph of the density function. At \(x=0\), \(f(x)\) is zero, and it increases, reaching a peak and then gradually decreasing towards zero as \(x\to\infty\). The shape of the graph depends on these parameters. It will have a right-skewed peak around the mode.

03

Calculating Probability for Exceeding 175

To find \(P(X > 175)\), use the cumulative distribution function (cdf):\[P(X > 175) = 1 - F(175) = 1 - \left[1 - e^{-(175/180)^9}\right] = e^{-(175/180)^9}.\]Compute this value for the probability.

04

Calculating Probability Between 150 and 175

To find \(P(150 < X < 175)\), calculate:\[P(150 < X < 175) = F(175) - F(150) = \left[1 - e^{-(175/180)^9}\right] - \left[1 - e^{-(150/180)^9}\right].\]Compute these values and find the probability difference.

05

Probability for Independent Specimens

The probability that at least one has strength between 150 and 175 is:\[1 - P(\text{both not in } (150, 175)) = 1 - (1 - P(150 < X < 175))^2.\]Substitute values from Step 4 and solve.

06

Finding the 10th Percentile Value

To find the strength value \(x\) such that \(P(X < x) = 0.1\), use:\[F(x) = 1 - e^{-(x/180)^9} = 0.1.\]Solve for \(x\):\[e^{-(x/180)^9} = 0.9,\]and take the logarithm:\[-(x/180)^9 = \ln(0.9),(x/180)^9 = -\ln(0.9),x = 180(-\ln(0.9))^{1/9}.\]Calculate \(x\).

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Key Concepts

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

Probability Density Function

The probability density function (pdf) is a fundamental concept in probability and statistics. It describes the likelihood of a random variable taking on a specific value. For the Weibull distribution, the pdf is defined as:

• \( f(x; \alpha, \beta) = \frac{\alpha}{\beta} \left(\frac{x}{\beta}\right)^{\alpha - 1} e^{-(x/\beta)^\alpha} \)
• The parameters \(\alpha\) and \(\beta\) shape the distribution's curve.
• \( \alpha \) is called the shape parameter, guiding the form of the distribution.
• \( \beta \) is the scale parameter, affecting the spread of the data.

The function is defined for \( x \geq 0 \), where \( x \) represents possible values such as tensile strength in this exercise. For this problem, with \(\alpha=9\) and \(\beta=180\), the pdf reaches a peak and then gradually decreases. This represents the probability of specific tensile strengths occurring in the composite material. It's crucial to observe how this curve behaves: it starts from zero, increases to a peak value, and finally tapers off approaching zero as \( x \rightarrow \infty \).

Cumulative Distribution Function

While the probability density function helps us understand the likelihood for individual data points, the cumulative distribution function (cdf) gives the probability that the random variable is less than or equal to a certain value. In a Weibull distribution, the cdf is formulated as:

• \( F(x; \alpha, \beta) = 1 - e^{-(x/\beta)^\alpha} \)

For example, if we want to find the probability that a specimen's strength is less than 175 MPa, the cdf is calculated as:

• \( F(175) = 1 - e^{-(175/180)^9} \)

Using this, we can also calculate probabilities between two points by finding the difference: \( P(150 < X < 175) = F(175) - F(150) \). This calculation tells us the proportion of specimens expected to have strengths within this range. By understanding both the pdf and cdf, one can effectively analyze and predict outcomes for continuous data distributions, especially in applications like material strength assessment.

Percentile Calculation

Percentiles are a useful aspect of statistics, allowing us to determine values below which a certain percentage of data falls. For the Weibull distribution, to find the percentile value \( x \) corresponding to the weakest 10% of specimens (the 10th percentile), we solve:

• \( F(x) = 0.1 \)

This translates to:

• \( 1 - e^{-(x/180)^9} = 0.1 \)

Then, by rearranging and solving for \( x \):

• \( e^{-(x/180)^9} = 0.9 \)
• \( (x/180)^9 = -\ln(0.9) \)
• \( x = 180(-\ln(0.9))^{1/9} \)

Calculating this gives us the threshold strength below which 10% of specimens lie. Percentile calculations are an essential tool in quality control and risk management, guiding decisions on acceptable performance criteria for products and systems.