The lifespan, expressed in years, of a motor for automating a gate manufactured by company A is a random variable $X$ that follows an exponential distribution with parameter $\lambda$, where $\lambda$ is a strictly positive real number. We know that $P ( X \leqslant 2 ) = 0.15$. Determine the exact value of the real number $\lambda$.
In the rest of the exercise, we will use 0.081 as the value of $\lambda$.
2. a. Determine $P ( X \geqslant 3 )$. b. Show that for all positive real numbers $t$ and $h$, $P _ { X \geqslant t } ( X \geqslant t + h ) = P ( X \geqslant h )$. c. The motor has already operated for 3 years. What is the probability that it will continue to operate for 2 more years? d. Calculate the expected value of the random variable $X$ and give an interpretation of this result.
3. In the rest of this exercise, results should be given rounded to $10 ^ { - 3 }$.
Company A announces that the percentage of defective motors in production is equal to $1 \%$. To verify this claim, 800 motors are randomly selected. It is found that 15 motors are detected as defective. Does the result of this test call into question the announcement of company A? Justify. You may use a confidence interval.