Recall that if $T$ follows an exponential distribution with parameter $\lambda$ ($\lambda$ being a strictly positive real number) then for any positive real $a$, $P(T \leqslant a) = \int_{0}^{a} \lambda \mathrm{e}^{-\lambda x} \mathrm{~d}x$. a. Show that $P(T \geqslant a) = \mathrm{e}^{-\lambda a}$. b. Show that if $T$ follows an exponential distribution then for all positive real numbers $t$ and $a$ we have $$P_{T \geqslant t}(T \geqslant t + a) = P(T \geqslant a).$$
In this part, the lifetime in hours of a light bulb without a defect is a random variable $T$ that follows the exponential distribution with expectation 10000. a. Determine the exact value of the parameter $\lambda$ of this distribution. b. Calculate the probability $P(T \geqslant 5000)$. c. Given that a light bulb without a defect has already operated for 7000 hours, calculate the probability that its total lifetime exceeds 12000 hours.
\begin{enumerate}
\item Recall that if $T$ follows an exponential distribution with parameter $\lambda$ ($\lambda$ being a strictly positive real number) then for any positive real $a$, $P(T \leqslant a) = \int_{0}^{a} \lambda \mathrm{e}^{-\lambda x} \mathrm{~d}x$.\\
a. Show that $P(T \geqslant a) = \mathrm{e}^{-\lambda a}$.\\
b. Show that if $T$ follows an exponential distribution then for all positive real numbers $t$ and $a$ we have
$$P_{T \geqslant t}(T \geqslant t + a) = P(T \geqslant a).$$
\item In this part, the lifetime in hours of a light bulb without a defect is a random variable $T$ that follows the exponential distribution with expectation 10000.\\
a. Determine the exact value of the parameter $\lambda$ of this distribution.\\
b. Calculate the probability $P(T \geqslant 5000)$.\\
c. Given that a light bulb without a defect has already operated for 7000 hours, calculate the probability that its total lifetime exceeds 12000 hours.
\end{enumerate}