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The length of time to failure (in hundreds of hours) for a
Chapter 4, Problem 12E(choose chapter or problem)
The length of time to failure (in hundreds of hours) for a transistor is a random variable Y with distribution function given by
\(F(y)=\left\{\begin{array}{ll} 0, & y<0, \\ 1-e^{-y^{2}}, & y \geq 0 . \end{array}\right.\)
a Show that \(F(y)\) has the properties of a distribution function.
b Find the .30-quantile, \(\phi_{.30}\), of Y.
c Find \(f(y)\).
d Find the probability that the transistor operates for at least 200 hours.
e Find \(P(Y>100 \mid Y \leq 200)\).
Questions & Answers
(1 Reviews)
QUESTION:
The length of time to failure (in hundreds of hours) for a transistor is a random variable Y with distribution function given by
\(F(y)=\left\{\begin{array}{ll} 0, & y<0, \\ 1-e^{-y^{2}}, & y \geq 0 . \end{array}\right.\)
a Show that \(F(y)\) has the properties of a distribution function.
b Find the .30-quantile, \(\phi_{.30}\), of Y.
c Find \(f(y)\).
d Find the probability that the transistor operates for at least 200 hours.
e Find \(P(Y>100 \mid Y \leq 200)\).
Step 1 of 6
(a)
The length of time to failure for a transistor(in hundreds of hours) is a random variable Y with a distribution function given by,
\(\begin{array}{l}F(y)=0\quad\ \ \ \ \ \ y<0\\ =1-e^{-y^2},\ y\ge0\end{array}\)
We need to show that F(y) has the properties of the distribution function.
The properties of distribution function If F(y) is a distribution function, then
1. \(F(-\infty)=\lim _{y \rightarrow-\infty} F(y)=0\)
2. \(F(\infty)=\lim _{y \rightarrow \infty} F(y)=1\)
3. F(y) is a non-decreasing function of y. [if \(y_{1}\) and \(y_{2}\) are any values such that \(y_{1}<y_{2}\) then \(F\left(y_{1}\right) \leq F\left(y_{2}\right)\)
\(F(0)=\lim _{y \rightarrow 0}\left(1-e^{-y^{2}}\right)=\left(1-e^{-0}\right)=0\)
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