A First Course in Probability (10th Edition)
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ISBN: 9780134753119
Author: Sheldon Ross
Publisher: PEARSON
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![Once an individual has been infected with a certain disease, let \( X \) represent the time (days) that elapses before the individual becomes infectious. An article proposes a Weibull distribution with \( \alpha = 2.6 \), \( \beta = 1.2 \), and \( \gamma = 0.5 \). [Hint: The two-parameter Weibull distribution can be generalized by introducing a third parameter \( \gamma \), called a threshold or location parameter: replace \( x \) in the equation below,
\[
f(x; \, \alpha, \, \beta) =
\begin{cases}
\frac{\alpha}{\beta^\alpha}x^{\alpha - 1}e^{-(x/\beta)^\alpha} & x \geq 0 \\
0 & x < 0
\end{cases}
\]
by \( x - \gamma \) and \( x \geq 0 \) by \( x \geq \gamma \).]
(a) Calculate \( P(1 < X < 2) \). (Round your answer to four decimal places.)
\[ \boxed{\phantom{1234}} \]
(b) Calculate \( P(X > 1.5) \). (Round your answer to four decimal places.)
\[ \boxed{\phantom{1234}} \]
(c) What is the 90th percentile of the distribution? (Round your answer to three decimal places.)
\(\boxed{\phantom{1234}}\) days
(d) What are the mean and standard deviation of \( X \)? (Round your answers to three decimal places.)
Mean: \(\boxed{\phantom{1234}}\) days
Standard Deviation: \(\boxed{\phantom{1234}}\) days](https://content.bartleby.com/qna-images/question/324f16c4-c4e5-4b06-b074-f261b39e021a/24a8314c-bd07-4d4b-90a7-2ce405af998f/1tj5u7d_processed.png)
Transcribed Image Text:Once an individual has been infected with a certain disease, let \( X \) represent the time (days) that elapses before the individual becomes infectious. An article proposes a Weibull distribution with \( \alpha = 2.6 \), \( \beta = 1.2 \), and \( \gamma = 0.5 \). [Hint: The two-parameter Weibull distribution can be generalized by introducing a third parameter \( \gamma \), called a threshold or location parameter: replace \( x \) in the equation below,
\[
f(x; \, \alpha, \, \beta) =
\begin{cases}
\frac{\alpha}{\beta^\alpha}x^{\alpha - 1}e^{-(x/\beta)^\alpha} & x \geq 0 \\
0 & x < 0
\end{cases}
\]
by \( x - \gamma \) and \( x \geq 0 \) by \( x \geq \gamma \).]
(a) Calculate \( P(1 < X < 2) \). (Round your answer to four decimal places.)
\[ \boxed{\phantom{1234}} \]
(b) Calculate \( P(X > 1.5) \). (Round your answer to four decimal places.)
\[ \boxed{\phantom{1234}} \]
(c) What is the 90th percentile of the distribution? (Round your answer to three decimal places.)
\(\boxed{\phantom{1234}}\) days
(d) What are the mean and standard deviation of \( X \)? (Round your answers to three decimal places.)
Mean: \(\boxed{\phantom{1234}}\) days
Standard Deviation: \(\boxed{\phantom{1234}}\) days
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