The length of time to failure (in hundreds of hours) for a

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 \(F\left(y_{1}\right) \leq F\left(y_{2}\right)\)

\(F(0)=\lim _{y \rightarrow 0}\left(1-e^{-y^{2}}\right)=\left(1-e^{-0}\right)=0\)