2. Let $Y$ denote the random variable which, for a small jar randomly selected from the production of line $\mathrm { F } _ { 2 }$, associates its sugar content. We assume that $Y$ follows the normal distribution with mean $m _ { 2 } = 0.17$ and standard deviation $\sigma _ { 2 }$. We further assume that the probability that a small jar randomly selected from the production of line $\mathrm { F } _ { 2 }$ is compliant equals 0.99 . Let Z be the random variable defined by $Z = \frac { Y - m _ { 2 } } { \sigma _ { 2 } }$. a. What distribution does the random variable $Z$ follow? b. Determine, as a function of $\sigma _ { 2 }$, the interval to which $Z$ belongs when $Y$ belongs to the interval $[ 0.16 ; 0.18 ]$. c. Deduce an approximate value to $10 ^ { - 3 }$ near of $\sigma _ { 2 }$.
You may use the table given below, in which the random variable $Z$ follows the normal distribution with mean 0 and standard deviation 1 .
| $\beta$ | $P ( - \beta \leqslant Z \leqslant \beta )$ |
| 2.4324 | 0.985 |
| 2.4573 | 0.986 |
| 2.4838 | 0.987 |
| 2.5121 | 0.988 |
| 2.5427 | 0.989 |
| 2.5758 | 0.990 |
| 2.6121 | 0.991 |
| 2.6521 | 0.992 |
| 2.6968 | 0.993 |
Exercise 3
Common to all candidates
Given a real number $k$, we consider the function $f _ { k }$ defined on $\mathbb { R }$ by
$$f _ { k } ( x ) = \frac { 1 } { 1 + \mathrm { e } ^ { - k x } }$$
The plane is equipped with an orthonormal coordinate system ( $\mathrm { O } , \vec { \imath } , \vec { \jmath }$ ).
Part A
In this part we choose $k = 1$. We have therefore, for all real $x , f _ { 1 } ( x ) = \frac { 1 } { 1 + \mathrm { e } ^ { - x } }$. The graph $\mathscr { C } _ { 1 }$ of the function $f _ { 1 }$ in the coordinate system ( $\mathrm { O } , \vec { \imath } , \vec { \jmath }$ ) is given in APPENDIX, to be returned with your answer sheet.
- Determine the limits of $f _ { 1 } ( x )$ as $x \to + \infty$ and as $x \to - \infty$ and give a graphical interpretation of the results obtained.
- Prove that, for all real $x , f _ { 1 } ( x ) = \frac { \mathrm { e } ^ { x } } { 1 + \mathrm { e } ^ { x } }$.
- Let $f _ { 1 } ^ { \prime }$ denote the derivative function of $f _ { 1 }$ on $\mathbb { R }$. Calculate, for all real $x , f _ { 1 } ^ { \prime } ( x )$.
Deduce the variations of the function $f _ { 1 }$ on $\mathbb { R }$.