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 a = 2.2 B = 1.3, and y = 0.5. [Hint: The two-parameter Weibull distribution can be generalized by introducing a third parameter y, called a threshold or location parameter: replace x in the equation below, x 2 0 (x; a, B) = x" - le-(x/B)a x < 0 by x - y and x 2 0 by x 2 y.] (a) Calculate P(1 < X < 2). (Round your answer to four decimal places.) (b) Calculate P(X > 1.5). (Round your answer to four decimal places.) (c) What is the 90th percentile of the distribution? (Round your answer to three decimal places.) days (d) What are the mean and standard deviation of X? (Round your answers to three decimal places.) mean days standard deviation days
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 a = 2.2 B = 1.3, and y = 0.5. [Hint: The two-parameter Weibull distribution can be generalized by introducing a third parameter y, called a threshold or location parameter: replace x in the equation below, x 2 0 (x; a, B) = x" - le-(x/B)a x < 0 by x - y and x 2 0 by x 2 y.] (a) Calculate P(1 < X < 2). (Round your answer to four decimal places.) (b) Calculate P(X > 1.5). (Round your answer to four decimal places.) (c) What is the 90th percentile of the distribution? (Round your answer to three decimal places.) days (d) What are the mean and standard deviation of X? (Round your answers to three decimal places.) mean days standard deviation days
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![**Weibull Distribution in Infectious Disease Modeling**
Once an individual has been infected with a certain disease, let \( X \) represent the time (in days) that elapses before the individual becomes infectious. An article proposes a Weibull distribution with \( \alpha = 2.2 \), \( \beta = 1.3 \), 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,
\[
f(x; \alpha, \beta) = \begin{cases}
\frac{\alpha}{\beta^\alpha} (x - \gamma)^{\alpha - 1} e^{-(x/\beta)^\alpha} & x \geq 0 \\
0 & x < 0
\end{cases}
\]
by \( x - \gamma \) and \( x \geq 0 \) by \( x \geq \gamma \).
(a) **Calculate \( P(1 < X < 2) \).** (Round your answer to four decimal places.)
\[
\text{Answer box:} \, \square
\]
(b) **Calculate \( P(X > 1.5) \).** (Round your answer to four decimal places.)
\[
\text{Answer box:} \, \square
\]
(c) **What is the 90th percentile of the distribution?** (Round your answer to three decimal places.)
\[
\text{Answer box:} \, \square \, \text{days}
\]
(d) **What are the mean and standard deviation of \( X \)?** (Round your answers to three decimal places.)
\[
\text{Mean:} \, \square \, \text{days}
\]
\[
\text{Standard deviation:} \, \square \, \text{days}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faf2a9245-87db-4e54-b147-768ccba693e7%2F4fee2551-e65a-49da-80e2-d8b6ee7f7c05%2Fj39j2g_processed.png&w=3840&q=75)
Transcribed Image Text:**Weibull Distribution in Infectious Disease Modeling**
Once an individual has been infected with a certain disease, let \( X \) represent the time (in days) that elapses before the individual becomes infectious. An article proposes a Weibull distribution with \( \alpha = 2.2 \), \( \beta = 1.3 \), 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,
\[
f(x; \alpha, \beta) = \begin{cases}
\frac{\alpha}{\beta^\alpha} (x - \gamma)^{\alpha - 1} e^{-(x/\beta)^\alpha} & x \geq 0 \\
0 & x < 0
\end{cases}
\]
by \( x - \gamma \) and \( x \geq 0 \) by \( x \geq \gamma \).
(a) **Calculate \( P(1 < X < 2) \).** (Round your answer to four decimal places.)
\[
\text{Answer box:} \, \square
\]
(b) **Calculate \( P(X > 1.5) \).** (Round your answer to four decimal places.)
\[
\text{Answer box:} \, \square
\]
(c) **What is the 90th percentile of the distribution?** (Round your answer to three decimal places.)
\[
\text{Answer box:} \, \square \, \text{days}
\]
(d) **What are the mean and standard deviation of \( X \)?** (Round your answers to three decimal places.)
\[
\text{Mean:} \, \square \, \text{days}
\]
\[
\text{Standard deviation:} \, \square \, \text{days}
\]
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