The lifetime of a valve, expressed in hours, is a random variable $T$ that follows the exponential distribution with parameter $\lambda = 0.0002$.

1. What is the average lifetime of a valve?
2. Calculate the probability, to 0.001, that the lifetime of a valve exceeds 6000 hours.

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.

In this exercise, we are interested in the operating mode of two restaurants: without reservation or with prior reservation.

1. The first restaurant operates without reservation but the waiting time to obtain a table is often a problem for customers. We model this waiting time in minutes by a random variable $X$ which follows an exponential distribution with parameter $\lambda$ where $\lambda$ is a strictly positive real number. We recall that the mathematical expectation of $X$ is equal to $\frac{1}{\lambda}$.
   A statistical study made it possible to observe that the average waiting time to obtain a table is 10 minutes. a. Determine the value of $\lambda$. b. What is the probability that a customer waits between 10 and 20 minutes to obtain a table? Round to $10^{-4}$. c. A customer has been waiting for 10 minutes. What is the probability that he must wait at least 5 more minutes to obtain a table? Round to $10^{-4}$.
2. The second restaurant has a capacity of 70 seats and only serves people who have made a prior reservation. The probability that a person who has made a reservation appears at the restaurant is estimated at 0.8. We denote by $n$ the number of reservations taken by the restaurant and $Y$ the random variable corresponding to the number of people who have made a reservation and appear at the restaurant. We admit that the behaviours of people who have made a reservation are independent of each other. The random variable $Y$ then follows a binomial distribution. a. Specify, as a function of $n$, the parameters of the distribution of the random variable $Y$, its mathematical expectation $E(Y)$ and its standard deviation $\sigma(Y)$. b. In this question, we denote by $Z$ a random variable following the normal distribution $\mathscr{N}\left(\mu, \sigma^{2}\right)$ with mean $\mu = 64.8$ and standard deviation $\sigma = 3.6$. Calculate the probability $p_{1}$ of the event $\{Z \leqslant 71\}$ using a calculator. c. We admit that when $n = 81$, $p_{1}$ is an approximate value to within $10^{-2}$ of the probability $p(Y \leqslant 70)$ of the event $\{Z \leqslant 70\}$. The restaurant has received 81 reservations. What is the probability that it cannot accommodate some of the customers who have made a reservation and appear?

Question 3

This hypermarket sells televisions whose lifespan, expressed in years, can be modelled by a random variable that follows an exponential distribution with parameter $\lambda$. The average lifespan of a television is eight years, which is expressed as: $\lambda = \frac{1}{8}$. The probability that a television chosen at random still works after six years has a value rounded to the nearest thousandth of: a. 0.750 b. 0.250 c. 0.472 d. 0.528

Part C

The lifetime (expressed in hours) of the electronic panel displaying the competitors' scores is a random variable $T$ that follows the exponential distribution with parameter $\lambda = 10^{-4}$ (expressed in $\mathrm{h}^{-1}$).

1. What is the probability that the panel functions for at least 2000 hours?
2. Organized presentation of knowledge

In this question, $\lambda$ denotes a strictly positive real number. Recall that the mathematical expectation of the random variable $T$ following an exponential distribution with parameter $\lambda$ is defined by: $\mathrm{E}(T) = \lim_{x \rightarrow +\infty} \int_{0}^{x} \lambda t \mathrm{e}^{-\lambda t} \mathrm{~d}t$. a. Consider the function $F$, defined for all real $t$ by: $F(t) = \left(-t - \frac{1}{\lambda}\right) \mathrm{e}^{-\lambda t}$.
Prove that the function $F$ is an antiderivative on $\mathbb{R}$ of the function $f$ defined for all real $t$ by: $f(t) = \lambda t \mathrm{e}^{-\lambda t}$. b. Deduce that the mathematical expectation of the random variable $T$ is equal to $\frac{1}{\lambda}$.
What is the expected lifetime of the electronic panel displaying the competitors' scores?

Let $X$ be a random variable that follows the exponential distribution with parameter $\lambda$, where $\lambda$ is a given strictly positive real number. We recall that the probability density of this distribution is the function $f$ defined on $[ 0 ; + \infty [$ by $$f ( x ) = \lambda \mathrm { e } ^ { - \lambda x } .$$
a. Let $c$ and $d$ be two real numbers such that $0 \leqslant c < d$.
Prove that the probability $P ( c \leqslant X \leqslant d )$ satisfies $$P ( c \leqslant X \leqslant d ) = \mathrm { e } ^ { - \lambda c } - \mathrm { e } ^ { - \lambda d } .$$
b. Determine a value of $\lambda$ to $10 ^ { - 3 }$ near such that the probability $P ( X > 20 )$ is equal to 0.05. c. Give the expectation of the random variable $X$
In the rest of the exercise we take $\boldsymbol { \lambda } = \mathbf { 0 , 1 5 }$.
d. Calculate $P ( 10 \leqslant X \leqslant 20 )$. e. Calculate the probability of the event $( X > 18 )$.
2. Let $Y$ be a random variable that follows the normal distribution with expectation 16 and standard deviation 1.95. a. Calculate the probability of the event $( 20 \leqslant Y \leqslant 21 )$. b. Calculate the probability of the event $( Y < 11 ) \cup ( Y > 21 )$.

Part A
We consider a random variable $X$ that follows the exponential distribution with parameter $\lambda$ where $\lambda > 0$. We recall that, for every strictly positive real number $a$, $$P(X \leqslant a) = \int_0^a \lambda \mathrm{e}^{-\lambda t} \,\mathrm{d}t$$ We propose to calculate the mathematical expectation of $X$, denoted $E(X)$, and defined by $$E(X) = \lim_{x \to +\infty} \int_0^x \lambda t \mathrm{e}^{-\lambda t} \,\mathrm{d}t$$ We admit that the function $F$ defined on $\mathbb{R}$ by $F(t) = -\left(t + \frac{1}{\lambda}\right)\mathrm{e}^{-\lambda t}$ is an antiderivative on $\mathbb{R}$ of the function $f$ defined on $\mathbb{R}$ by $f(t) = \lambda t \mathrm{e}^{-\lambda t}$.

1. Let $x$ be a strictly positive real number. Verify that $$\int_0^x \lambda t \mathrm{e}^{-\lambda t} \,\mathrm{d}t = \frac{1}{\lambda}\left(-\lambda x \mathrm{e}^{-\lambda x} - \mathrm{e}^{-\lambda x} + 1\right)$$
2. Deduce that $E(X) = \frac{1}{\lambda}$.

Part B
The lifetime, expressed in years, of an electronic component can be modeled by a random variable denoted $X$ following the exponential distribution with parameter $\lambda$ where $\lambda > 0$.

1. On the graph of appendix 2 (to be returned with the answer sheet): a. Represent the probability $P(X \leqslant 1)$. b. Indicate where the value of $\lambda$ can be read directly.
2. We assume that $E(X) = 2$. a. What does the value of the mathematical expectation of the random variable $X$ represent in the context of the exercise? b. Calculate the value of $\lambda$. c. Calculate $P(X \leqslant 2)$. Give the exact value then the value rounded to 0.01. Interpret this result. d. Given that the component has already functioned for one year, what is the probability that its total lifetime is at least three years? Give the exact value.

Part C
An electronic circuit is composed of two identical components numbered 1 and 2. We denote by $D_1$ the event ``component 1 fails before one year'' and we denote by $D_2$ the event ``component 2 fails before one year''. We assume that the two events $D_1$ and $D_2$ are independent and that $P(D_1) = P(D_2) = 0,39$. Two possible configurations are considered:

1. When the two components are connected ``in parallel'', circuit A fails only if both components fail at the same time. Calculate the probability that circuit A fails before one year.
2. When the two components are connected ``in series'', circuit B fails as soon as at least one of the two components fails. Calculate the probability that circuit B fails before one year.

Cesium 137 is a radioactive element that constitutes one of the main sources of radioactivity in nuclear reactor waste. The time $T$, in years, during which a cesium 137 atom remains radioactive can be approximated by a random variable $T$ that follows the exponential distribution with parameter $\lambda = \frac { \ln 2 } { 30 }$. What is the probability that a cesium 137 atom remains radioactive for at least 60 years? a. 0.125 b. 0.25 c. 0.75 d. 0.875

1. 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 $$P_{T \geqslant t}(T \geqslant t + a) = P(T \geqslant a).$$
2. 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.

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 $a \geqslant 0, P(T \leqslant a) = \int_{0}^{a} f(x) \mathrm{d}x$.

1. 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 $\lim_{t \rightarrow +\infty} P(T \leqslant t) = 1$.
2. Suppose that $P(T \leqslant 7) = 0.5$. Determine $\lambda$ to the nearest $10^{-3}$.
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.

Exercise 3

Part A

An astronomer responsible for an astronomy club observed the sky one August evening in 2015 to see shooting stars. He made observations of the waiting time between two appearances of shooting stars. He then modelled this waiting time, expressed in minutes, by a random variable $T$ which follows an exponential distribution with parameter $\lambda$. By exploiting the data obtained, he established that $\lambda = 0,2$.

1. When the group sees a shooting star, verify that the probability that it waits less than 3 minutes to see the next shooting star is approximately 0,451.
2. When the group sees a shooting star, what minimum duration must it wait to see the next one with a probability greater than 0,95? Round this time to the nearest minute.
3. The astronomer has planned an outing of two hours. Estimate the average number of observations of shooting stars during this outing.

Part B

This manager sends a questionnaire to his members to get to know them better. He obtains the following information:

• $64 \%$ of the people surveyed are new members;
• $27 \%$ of the people surveyed are former members who own a personal telescope;
• $65 \%$ of new members do not have a personal telescope.

1. A member is chosen at random. Show that the probability that this member owns a personal telescope is 0,494.
2. A member is chosen at random from among those who own a personal telescope. What is the probability that this is a new member? Round to $10 ^ { - 3 }$ near.

Part C

For practical reasons, the astronomer responsible for the club would like to install an observation site on the heights of a small town of 2500 inhabitants. But light pollution due to public lighting harms the quality of observations. To try to convince the town hall to cut off the night lighting during observation nights, the astronomer conducts a random survey of 100 inhabitants and obtains 54 favourable opinions on cutting off the night lighting. The astronomer makes the hypothesis that $50 \%$ of the village population is in favour of cutting off the night lighting. Does the result of this survey lead him to change his mind?

Part A - Waiting time

1. In this question, we are interested in the waiting time of an Internet customer when they contact the telephone assistance before reaching an operator. A study makes it possible to model this waiting time in minutes by the random variable $D_1$ which follows the exponential distribution with parameter 0.6. a. What is the average waiting time that an Internet customer calling this assistance line can expect? b. Calculate the probability that the waiting time of a randomly chosen Internet customer is less than 5 minutes.
2. In this question, we are interested in the waiting time of a mobile customer when they contact the telephone assistance before reaching an operator. We model this waiting time in minutes by the random variable $D_2$ which follows an exponential distribution with parameter $\lambda$, $\lambda$ being a strictly positive real number. a. Given that $P\left(D_2 \leqslant 4\right) = 0.798$, determine the value of $\lambda$. b. Taking $\lambda = 0.4$, can we consider that fewer than $10\%$ of randomly chosen mobile customers wait more than 5 minutes before reaching an operator?

In this exercise, we study some characteristic quantities of the operation of parking lots in a city. Throughout the exercise, probabilities will be given with a precision of $10 ^ { - 4 }$.

Parts A, B, and C are independent

Part A - Waiting time to enter an underground parking lot

The waiting time is defined as the time that elapses between the moment the car arrives at the parking entrance and the moment it passes through the parking entrance barrier. The following table presents observations made over one day.

Waiting time in minutes	$[ 0 ; 2 [$	$[ 2 ; 4 [$	$[ 4 ; 6 [$	$[ 6 ; 8 [$
Number of cars	75	19	10	5

1. Propose an estimate of the average waiting time for a car at the parking entrance.
2. We decide to model this waiting time by a random variable $T$ following an exponential distribution with parameter $\lambda$ (expressed in minutes). a. Justify that we can choose $\lambda = 0.5 \mathrm {~min}$. b. A car arrives at the parking entrance. What is the probability that it takes less than two minutes to pass through the barrier? c. A car has been waiting at the parking entrance for one minute. What is the probability that it passes through the barrier in the next minute?

Part B - Duration and parking rates in this underground parking lot

Once parked, the parking duration of a car is modeled by a random variable $D$ that follows a normal distribution with mean $\mu = 70 \mathrm {~min}$ and standard deviation $\sigma = 30 \mathrm {~min}$.

1. a. What is the average parking duration for a car? b. A motorist enters and parks in the parking lot. What is the probability that their parking duration exceeds two hours? c. To the nearest minute, what is the maximum parking time for at least $99 \%$ of cars?
2. The parking duration is limited to three hours. The table gives the rate for the first hour and each additional hour is charged at a single rate. Any hour started is charged in full.

\begin{tabular}{ c } Parking

duration

& Less than 15 min & Between 15 min and 1 h &

Additional

hour

\hline Rate in euros & Free & 3.5 & $t$ \hline \end{tabular}
Determine the rate $t$ for the additional hour that the parking manager must set so that the average parking price for a car is 5 euros.

Part C - Waiting time to park in a city center parking lot

The parking duration of a car in a city center parking lot is modeled by a random variable $T ^ { \prime }$ that follows a normal distribution with mean $\mu ^ { \prime }$ and standard deviation $\sigma ^ { \prime }$. It is known that the average parking time in this lot is 30 minutes and that $75 \%$ of cars have a parking time less than 37 minutes. The parking manager aims for the objective that $95 \%$ of cars have a parking time between 10 and 50 minutes. Is this objective achieved?

The operating lifetime, expressed in days, of a machine used for coating, is modelled by a random variable $Y$ which follows the exponential distribution with mean equal to 500 days.
What is the probability, rounded to the nearest hundredth, that the operating lifetime of the machine is less than or equal to 300 days?
Answer a: 0.45 Answer b: 1 Answer c: 0.55 Answer d: We cannot answer because data is missing

A type of oscilloscope has a lifespan, expressed in years, which can be modelled by a random variable $D$ that follows an exponential distribution with parameter $\lambda$. It is known that the average lifespan of this type of oscilloscope is 8 years. Statement 1: for an oscilloscope of this type chosen at random and having already operated for 3 years, the probability that the lifespan is greater than or equal to 10 years, rounded to the nearest hundredth, is equal to 0.42. Recall that if $X$ is a random variable that follows an exponential distribution with parameter $\lambda$, then for all positive real $t$: $P(X \leqslant t) = 1 - \mathrm{e}^{-\lambda t}$. Indicate whether Statement 1 is true or false, justifying your answer.

During the fifteen days preceding the start of the university term, the telephone switchboard of a student mutual aid organisation records a record number of calls. Callers are first placed on hold and hear background music and a pre-recorded message. During this first phase, the waiting time, expressed in seconds, is modelled by the random variable $X$ which follows the exponential distribution with parameter $\lambda = 0{,}02 \mathrm{~s}^{-1}$. Callers are then connected with a customer service representative who answers their questions. The exchange time, expressed in seconds, during this second phase is modelled by the random variable $Y$, expressed in seconds, which follows the normal distribution with mean $\mu = 96$ s and standard deviation $\sigma = 26 \mathrm{~s}$.

1. What is the average total duration of a call to the telephone switchboard (waiting time and exchange time with the customer service representative)?
2. A student is chosen at random from among the callers to the telephone switchboard. a. Calculate the probability that the student is placed on hold for more than 2 minutes. b. Calculate the probability that the exchange time with the adviser is less than 90 seconds.
3. A female student, chosen at random from among the callers, has been waiting for more than one minute to be connected with the customer service. Tired, she hangs up and dials the number again. She hopes to wait less than thirty seconds this time. Does hanging up and calling back increase her chances of limiting the additional waiting time to 30 seconds, or would she have been better off staying on the line?

Let $T$ denote the random variable equal to the lifespan, in months, of a stopwatch and we assume it follows an exponential distribution with parameter $\lambda = 0.0555$.

1. Calculate the average lifespan of a stopwatch (rounded to the nearest unit).
2. Calculate the probability that a stopwatch has a lifespan between one and two years.
3. A coach has not changed his stopwatch for two years. What is the probability that it will still be in working order for at least one more year?

5 points
The probabilities requested should be rounded to 0.01.
A shopkeeper has just equipped himself with an Italian ice cream dispenser.

1. The duration, in months, of operation without breakdown of his Italian ice cream dispenser is modelled by a random variable $X$ which follows an exponential distribution with parameter $\lambda$ where $\lambda$ is a strictly positive real number (recall that the density function $f$ of the exponential distribution is given on $\left[ 0 ; + \infty \left[ \text{ by } f ( x ) = \lambda \mathrm { e } ^ { - \lambda x } \right. \right.$).

The seller of the device assures that the average duration of operation without breakdown of this type of dispenser, that is to say the mathematical expectation of $X$, is 10 months. a. Justify that $\lambda = 0.1$. b. Calculate the probability that the Italian ice cream dispenser has experienced no breakdown during the first six months. c. Given that the dispenser has experienced no breakdown during the first six months, what is the probability that it experiences none until the end of the first year? Justify. d. The shopkeeper will replace his Italian ice cream dispenser after a time $t$, expressed in months, which verifies that the probability of the event ( $X > t$ ) is equal to 0.05. Determine the value of $t$ rounded to the nearest integer.
2. The dispenser manual specifies that the dispenser provides Italian ice creams whose mass is between 55 g and 65 g. Consider the random variable $M$ representing the mass, in grams, of a distributed ice cream. It is admitted that $M$ follows the normal distribution with expectation 60 and standard deviation 2.5. a. Calculate the probability that the mass of an Italian ice cream chosen at random from those distributed is between 55 g and 65 g. b. Determine the largest value of $m$, rounded to the nearest gram, such that the probability $P ( M \geqslant m )$ is greater than or equal to 0.99.
3. The Italian ice cream dispenser allows you to choose only one of two flavours: vanilla or strawberry. To better manage his purchases of raw materials, the shopkeeper makes the hypothesis that there will be in proportion two vanilla ice cream buyers for one strawberry ice cream buyer. On the first day of use of his dispenser, he observes that out of 120 consumers, 65 chose vanilla ice cream. For what mathematical reason could he doubt his hypothesis? Justify.

In a snowy corridor, the time interval separating two successive avalanches, called the occurrence time of an avalanche, expressed in years, is modelled by a random variable $T$ which follows an exponential distribution. It has been established that an avalanche is triggered on average every 5 years. Thus $E ( T ) = 5$. The probability $P ( T \geqslant 5 )$ is equal to: a. 0.5 b. $1 - \mathrm{e}^{-1}$ c. $\mathrm{e}^{-1}$ d. $\mathrm{e}^{-25}$

The duration, in days, of use of precision electronic scales before misalignment is modeled by a random variable $T$ which follows an exponential distribution with parameter $\lambda$. The representative curve of the density function of this random variable $T$ is given.

1. a. By graphical reading, give a bound for $\lambda$ with amplitude 0.01. b. The area of the shaded region, in square units, is equal to 0.45. Determine the exact value of $\lambda$.

In the following, we will take $\lambda = 0.054$.

1. Determine, to the nearest day, the average duration of use of a scale without it becoming misaligned.
2. A scale is put into service on January 1st, 2020. It operates without misalignment from January 1st to January 20 inclusive. Determine the probability that it operates without misalignment until January 31 inclusive.

The operating duration, expressed in years, of a motor until the first failure occurs is modelled by a random variable following an exponential distribution with parameter $\lambda$ where $\lambda$ is a strictly positive real number. The probability that the motor operates without failure for more than 3 years is equal to: Answer A: $e ^ { - 3 \lambda } \quad$ Answer B: $1 - e ^ { - 3 \lambda } \quad$ Answer C: $e ^ { 3 \lambda } - 1 \quad$ Answer D: $e ^ { 3 \lambda }$

Given that for each $a \in (0,1)$,
$$\lim_{h \rightarrow 0^+} \int_{h}^{1-h} t^{-a}(1-t)^{a-1}\, dt$$
exists. Let this limit be $g(a)$. In addition, it is given that the function $g(a)$ is differentiable on $(0,1)$.
The value of $g\left(\frac{1}{2}\right)$ is
(A) $\pi$
(B) $2\pi$
(C) $\frac{\pi}{2}$
(D) $\frac{\pi}{4}$

The value of $g'\left(\frac{1}{2}\right)$ is
(A) $\frac{\pi}{2}$
(B) $\pi$
(C) $-\frac{\pi}{2}$
(D) 0

(1) If the probability density function $f ( t )$ of a continuous random variable $T$ is denoted by
$$f ( t ) = \begin{cases} \lambda e ^ { - \lambda t } & ( t \geq 0 ) \\ 0 & ( t < 0 ) \end{cases}$$
with a positive constant $\lambda$, then we say that $T$ follows an exponential distribution with parameter $\lambda$. Compute the average of this random variable. Also, derive the probability distribution function $F ( t ) = P ( T \leq t )$ of this exponential distribution, where $P ( X )$ is the probability of the event $X$.
(2) Show that the probability distribution given in question (1) is memoryless. Namely, show that
$$P ( T > s + t \mid T > s ) = P ( T > t )$$
holds for any $s > 0$ and $t > 0$, where $P ( X \mid Y )$ is the probability of the event $X$ conditioned on the event $Y$.
(3) Let us call the time interval between the time when one starts solving a problem and the time when one finishes it "time required for solution". Assume that the time required for solution of each of $n$ students follows the exponential distribution with the same parameter $\lambda _ { 0 }$. Let all the $n$ students start solving the problem at the same time. Find the probability distribution function and the average of the time required for solution of the student who finishes solving the problem earliest. Here, the time required for solution of each student is mutually independent.
(4) Assume that the times required for solution of a student A and a student B follow the exponential distributions with parameters $\lambda _ { A }$ and $\lambda _ { B }$, respectively. Let the two students start solving the problem at the same time. Find the probability that student A finishes solving the problem earlier than student B.
(5) Let a smart student Hideo and other ten students start solving a problem at the same time. Assume that the time required for solution of each of all the students follows an exponential distribution, where the average time required for solution of each of all the students except Hideo is ten times longer than that of Hideo. Find the probabilities that Hideo finishes solving the problem first and fourth, respectively.

Consider a light whose state alternately and repeatedly switches between the OFF (no light) state and the ON (light) state. For each OFF and ON state, the duration, represented by $T _ { 0 }$ and $T _ { 1 }$ respectively, changes at each transition and is independent.
By using $t$, which represents elapsed time from the initiation of each state, $T _ { 0 }$ and $T _ { 1 }$ follow the exponential distribution whose probability density functions are described respectively as
$$f _ { 0 } ( t ) = \lambda _ { 0 } e ^ { - \lambda _ { 0 } t } \quad \left( \lambda _ { 0 } > 0 \right)$$
and
$$f _ { 1 } ( t ) = \lambda _ { 1 } e ^ { - \lambda _ { 1 } t } \quad \left( \lambda _ { 1 } > 0 \right)$$
Here, for example, $P _ { 0 } ( a , b )$, which is the probability that the condition $a \leq T _ { 0 } \leq b ( 0 \leq a \leq b )$ is satisfied, can be calculated as
$$P _ { 0 } ( a , b ) = \int _ { a } ^ { b } f _ { 0 } ( t ) \mathrm { d } t$$
Assume that the light switches from the ON state to the OFF state at time $\tau = 0$. Answer the following questions.
I. Calculate the expected value and the standard deviation of $T _ { 0 }$.
II. Calculate the expected value and the standard deviation of $T _ { 0 } + T _ { 1 }$.
III. Consider a situation where time tends towards infinity and the condition $\left( \lambda _ { 0 } + \lambda _ { 1 } \right) \tau \rightarrow \infty$ approximately holds.
1. Calculate the probability that the light is in the OFF state. 2. Calculate the expected value of the remaining time from the current state to the next state transition of the light.
IV. At the time $\tau = \tau _ { x } \left( \tau _ { x } > 0 \right)$, calculate the probability that the light is in the ON state for the first occasion after $\tau = 0$.