The lifetime, in years, of an electronic component manufactured in this factory is a random variable $T$ that follows the exponential distribution with parameter $\lambda$ (where $\lambda$ is a strictly positive real number). We denote by $f$ the density function associated with the random variable $T$. We recall that:
for every real number $x \geqslant 0, f(x) = \lambda \mathrm{e}^{-\lambda x}$.
for every real number $a \geqslant 0, P(T \leqslant a) = \int_{0}^{a} f(x) \mathrm{d}x$.
The representative curve $\mathscr{C}$ of the function $f$ is given. a. Give a graphical interpretation of $P(T \leqslant a)$ where $a > 0$. b. Show that for every real number $t \geqslant 0 : P(T \leqslant t) = 1 - \mathrm{e}^{-\lambda t}$. c. Deduce that $\lim_{t \rightarrow +\infty} P(T \leqslant t) = 1$.
Suppose that $P(T \leqslant 7) = 0.5$. Determine $\lambda$ to the nearest $10^{-3}$.
In this question we take $\lambda = 0.099$ and round the results of probabilities to the nearest hundredth. a. A component manufactured in this factory is chosen at random. Determine the probability that this component operates for at least 5 years. b. A component is chosen at random from among those that still function after 2 years. Determine the probability that this component has a lifetime greater than 7 years. c. Give the mathematical expectation $\mathrm{E}(T)$ of the random variable $T$ to the nearest unit. Interpret this result.
The lifetime, in years, of an electronic component manufactured in this factory is a random variable $T$ that follows the exponential distribution with parameter $\lambda$ (where $\lambda$ is a strictly positive real number). We denote by $f$ the density function associated with the random variable $T$. We recall that:
\begin{itemize}
\item for every real number $x \geqslant 0, f(x) = \lambda \mathrm{e}^{-\lambda x}$.
\item for every real number $a \geqslant 0, P(T \leqslant a) = \int_{0}^{a} f(x) \mathrm{d}x$.
\end{itemize}
\begin{enumerate}
\item The representative curve $\mathscr{C}$ of the function $f$ is given.\\
a. Give a graphical interpretation of $P(T \leqslant a)$ where $a > 0$.\\
b. Show that for every real number $t \geqslant 0 : P(T \leqslant t) = 1 - \mathrm{e}^{-\lambda t}$.\\
c. Deduce that $\lim_{t \rightarrow +\infty} P(T \leqslant t) = 1$.
\item Suppose that $P(T \leqslant 7) = 0.5$. Determine $\lambda$ to the nearest $10^{-3}$.
\item In this question we take $\lambda = 0.099$ and round the results of probabilities to the nearest hundredth.\\
a. A component manufactured in this factory is chosen at random. Determine the probability that this component operates for at least 5 years.\\
b. A component is chosen at random from among those that still function after 2 years. Determine the probability that this component has a lifetime greater than 7 years.\\
c. Give the mathematical expectation $\mathrm{E}(T)$ of the random variable $T$ to the nearest unit. Interpret this result.
\end{enumerate}