a) Suppose that a r.v. Y has Exponential Distribution Y-Exp(1). Find the distribution of W = BY for some ß> 0. ● ● Fy(y) = F (w) - Pr(W ≤w) =_ Definition: A random variable W is said to have the Weibull distribution with parameters B and k if its cdf is of the form Fw (w)=1-e b) Write down the pdf of W fw (w) = (Fw (w)) 0 for w> 0.. w>0 otherwise
a) Suppose that a r.v. Y has Exponential Distribution Y-Exp(1). Find the distribution of W = BY for some ß> 0. ● ● Fy(y) = F (w) - Pr(W ≤w) =_ Definition: A random variable W is said to have the Weibull distribution with parameters B and k if its cdf is of the form Fw (w)=1-e b) Write down the pdf of W fw (w) = (Fw (w)) 0 for w> 0.. w>0 otherwise
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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![**Problem Statement:**
a) Suppose that a random variable \( Y \) has an Exponential Distribution \( Y \sim \text{Exp}(1) \). Find the distribution of \( W = \beta \sqrt{Y} \) for some \( \beta > 0 \).
- \( F_Y(y) = \) ______________.
- \( F_W(w) = \Pr(W \leq w) = \) ______________.
**Definition:**
A random variable \( W \) is said to have the Weibull distribution with parameters \( \beta \) and \( k \) if its cumulative distribution function (cdf) is of the form:
\[
F_W(w) = 1 - e^{-\left(\frac{w}{\beta}\right)^k}, \quad \text{for } w > 0
\]
b) Write down the probability density function (pdf) of \( W \).
\[
f_W(w) = (F_W(w)) =
\begin{cases}
\quad \quad \_\_\_\_\_\_\_\_\_\_\_\_, & w > 0 \\
0, & \text{otherwise}
\end{cases}
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Transcribed Image Text:**Problem Statement:**
a) Suppose that a random variable \( Y \) has an Exponential Distribution \( Y \sim \text{Exp}(1) \). Find the distribution of \( W = \beta \sqrt{Y} \) for some \( \beta > 0 \).
- \( F_Y(y) = \) ______________.
- \( F_W(w) = \Pr(W \leq w) = \) ______________.
**Definition:**
A random variable \( W \) is said to have the Weibull distribution with parameters \( \beta \) and \( k \) if its cumulative distribution function (cdf) is of the form:
\[
F_W(w) = 1 - e^{-\left(\frac{w}{\beta}\right)^k}, \quad \text{for } w > 0
\]
b) Write down the probability density function (pdf) of \( W \).
\[
f_W(w) = (F_W(w)) =
\begin{cases}
\quad \quad \_\_\_\_\_\_\_\_\_\_\_\_, & w > 0 \\
0, & \text{otherwise}
\end{cases}
\]
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