Grade 12 - Comp. / Density laws 48 exercises (including 47 corrected)

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234I2JO 1. Exponential law E.4166 Let X be a continuous random variable that follows an exponential distribution with param-eter – . Recall that : P ( X a ) = a 0 – · e − λt d t The curve given below represents the associated density func-tion : 1 Interpret on the graph the probability P ( X 1) . 2 Indicate on the graph where the parameter – reads di-rectly. E.4171 We denote X a continuous ran-dom variable that follows an exponential law with parameter – =0.04 . Recall that for any positive real t , the probability of the event ( X t ) , denoted P ( X t ) , is given by: P ( X t ) = t 0 – · e − λ · x d x Determine the rounded value of P ( X > 5) to within 10 − 2 by excess. E.4159 The lifetime, expressed in hours, of an electronic game, is a random variable X which follows the exponential law with parameter – =0.000 3 . Recall that, for any t 0 : P ( X t ) = t 0 – · e − λ · x d x Indicate whether the following statement is true or false, jus-tifying the answer. Assertion: The probability that the duration of this game is strictly greater than 2 000 hours is less than 0.5 . E.4158 A random variable X follows the exponential law with parameter – ( –> 0 ) . Recall that for any real a> 0 : P ( X a )= a 0 – · e − λ · t d t Indicate whether the following proposition is true or false, and give a justification for the answer chosen. Proposition: The real a such that P ( X >a )= P ( X a ) is equal to ln 2 – . 2. Exponential law and expectation E.6958 An astronomer takes readings of the waiting time between two shooting star apparitions. He then models this waiting time, expressed in minutes, by a random variable T that follows an exponential law of param-eter – . By exploiting the data obtained, he established that – =0.2 . The astronomer plans to observe the sky for two hours. Es-timate the average number of shooting star sightings on this outing. 3. Exponential law and parameter search E.4190 Consider the random variable X following an exponential law with parameter – with –> 0 . Thus, we have probability: P ( X t )= t 0 – · e − λ · x d x Determine – , rounded to the nearest 10 − 1 , so that the prob-ability P ( X > 6) is equal to 0.3 . E.4148 Consider a random variable X on a probability space Ω ; P following an exponential law with parameter – . Recall that, for any positive real number k : P ( X k ) = k 0 – · e − λ · x d x 1 Using the previous formula, show that : P 500 X 1 000 = e − 500 · λ − e − 1000 · λ 2 In this question, any trace of research, however incom-plete, or initiative, however unsuccessful, will be taken into account in the assessment. A study decides to model the number of kilometers cov-ered by a tire without a puncture by a random variable X following an exponential distribution with parameter – . The probability that the tire will travel between 500 and 1 000 kilometers without a puncture being equal to 1 4 , determine the value rounded to 10 − 4 of the parameter – . 4. Exponential law and conditional probability https://chingmath.fr chapExoCorrec/4166 sacados/4166 Extrait de Antilles-Guyane Juin 2006 234I2JO chapExoCorrec/4171 sacados/4171 Extrait de Liban Juin 2009 chapExoCorrec/4159 sacados/4159 chapExoCorrec/4158 sacados/4158 Extrait de Liban Juin 2010 chapExoCorrec/6958 sacados/6958 chapExoCorrec/4190 sacados/4190 Extrait de Liban Mai 2006 chapExoCorrec/4148 sacados/4148
E.4187 Alain is an amateur electronics manufacturer. He buys components from a store that all ap-pear to be identical, but some of them are defective. The probability that a component sold in the store is defective is estimated to be 0.02 . We assume that the lifetime T 1 (in hours) of each defective component follows an exponential distribution with parame-ter – 1 =5 × 10 − 4 and that the lifetime T 2 (in hours) of each non-defective component follows an exponential distribution with parameter – 2 =10 − 4 (see the form below) . 1 Calculate the probability that the lifetime of a compo-nent will exceed 1 000 hours : a if this component is defective ; b if this component is not defective. Give a rounded value for these probabilities to 10 − 2 near. 2 Let T be the lifetime (in hours) of a randomly purchased component. Show that the probability that this component will still be in working order after t hours of operation is : P ( T t ) = 0.02 · e − 5 × 10 − 4 · t + 0.98 · e − 10 − 4 · t (remember that the probability that a component sold in the store is defective is equal to 0.02 ) 3 Given that the purchased component is still functioning 1 000 hours after installation, what is the probability that this component is defective? Give a value for this probability, rounded to the nearest 10 − 2 . 5. Exponential law and lifespan without aging E.4186 The waiting time T , in minutes, at a freeway tollbooth before the checkout is a random variable that follows an exponential law with parameter – = 1 6 . So for any real t> 0 : P ( X <t ) = t 0 – · e − λ · x d x avec – = 1 6 where t denotes time expressed in minutes. Knowing that a motorist has already waited 2 minutes, what is the probability (rounded to within 10 − 4 ) that his total time is less than 5 minutes? E.4154 An urn contains 10 white balls and n red balls, n being a natural number greater than or equal to 2 . A player is asked to draw balls from the urn. On each draw, all the balls have the same probability of being drawn. For each white ball drawn, he wins 2 euros and for each red ball drawn, he loses 3 euros. It is assumed that n =1000 . The urn therefore contains 10 white balls and 1000 red balls. The player is unaware that the game is completely unfavor-able to him and decides to make several draws without reset-ting until he gets a white ball. Since the number of white balls is small compared with the number of red balls, we admit that we can model the num-ber of draws necessary to obtain a white ball by a random variable Z following the law, for any k ∈ N : P ( Z k ) = k 0 0.01 · e − 0.01 · x d x The following questions will therefore be answered using this model : 1 Calculate the probability that the player will need to shoot at most 50 balls to get a white ball, i.e., P ( Z 50) . 2 Calculate the conditional probability of the event : ˇ the player drew at most 60 balls to draw a white ball ı know-ing that the event ˇ the player drew more than 50 balls to draw a white ball ı. E.4149 The lifetime, expressed in years of a device, is modeled by a random variable X that follows the exponential law of parameter – on 0 ; + ∞ Recall that for any t> 0 , the probability of the event ( X t ) is given by: P X t = t 0 – · e − λ · x d x (avec – =0 ; 07 ) . Without justification, indicate whether each of the following propositions is true or false. Proposition 1: the probability that the device has a lifetime greater than 10 years is equal to 0.5 to within 10 − 2 . Proposition 2: knowing that the device has worked 10 years, the probability that it will still work 10 years is equal to 0.5 to within 10 − 2 . E.6263 A restaurant operates without reservations, but the waiting time for a table is often a prob-lem for customers. We model this waiting time in minutes by a random variable X which follows an exponential law with parameter – where – is a strictly positive real. Recall that the mathematical expectation of X is equal to 1 – . A statistical study has shown that the average waiting time for a table is 10 minutes. 1 Determine the value of – . 2 What is the probability that a customer will wait between 10 and 20 minutes to get a table? We’ll round to 10 − 4 . 3 A customer has been waiting for 10 minutes. What is the probability that he will have to wait at least 5 minutes longer to get a table? We’ll round up to 10 − 4 . https://chingmath.fr chapExoCorrec/4187 sacados/4187 chapExoCorrec/4186 sacados/4186 chapExoCorrec/4154 sacados/4154 chapExoCorrec/4149 sacados/4149 chapExoCorrec/6263 sacados/6263
01234512Cf 1,21,41,61,822,22,4Nombre en milliers24Cf100plantsTaille enm E.4175 A store manager buys electronic components. The lifetime of one of these components is a random variable noted X which follows a law of life without aging or exponen-tial law of parameter – , with – real strictly positive. 1 Knowing that P ( X > 5)=0.325 , determine – , rounded to the nearest thousandth. For the following questions, we’ll take : – =0.225 2 What is the probability that a component will last less than 8 years? More than 8 years? 3 What is the probability that a component will last more than 8 years knowing that it has already lasted more than 3 years? 6. Introduction E.4208 Shown below is the representative curve of the function f defined on the interval 1 ; 5 by the rela-tion : f ( x ) = 32 (3 x + 1) 2 Consider a dart-throwing game based on the shaded surface defined by: the x-axis and the curve C f ; the straight lines with equations x =1 and x =5 . Assuming that on each throw, the dart falls into this shaded area, we wish to know the probability of the dart reaching the hatched area bounded by the two straight lines of equations x =3 and x =5 . To do this, consider the random variable X which associates with each dart thrown the abscissa of its point of reception on the target. 1 Determine the primitive of the function f . 2 Determine the values of the following two integrals : 5 1 f ( x ) d x ; 5 3 f ( x ) d x 3 Deduce the probability: P 3 X 5 E.4056 1 Let X be the random variable equiprobably choosing an integer in the interval 0 ; 10 . Determine probability: P X =5 2 Let X be the random variable equiprobably choosing an integer in the interval 0 ; 999 . Determine probability: P X =5 3 Let X be the random variable equiprobably choosing a real the interval 0 ; 10 . Determine probability: P X =5 4 Let X be the random variable equiprobably choosing a real in the interval 0 ; 10 . Determine probability: P X ∈ 0 ; 5 E.6407 A statistical study focuses on the size of corn plants : All approximate values requested in this exercise will be rounded to the nearest hundredth. 1 From a histogram: As indicated by the legend of the histogram, we can use the graduation of the y-axis (required for the curve C f ) to de-termine the number associated with a bar : 0.1 × 1 000 = 100 plants The readings were used to create the histogram above. a How many plants between 1.9 m and 2.2 m are in-cluded in this study? b If we choose a plant at random from among the plants in this study, what is the probability that this plant will be between 1.9 m and 2.2 m in size? We will round this probability to two decimal places. 2 From the curve: We choose to use a curve that ˇroughlyı represents the histogram. The expression of this function is : f ( x ) = 15 · 4 · x − 5 12 · x 2 − 30 · x + 22 a Determine a primitive of the function f . b Determine the value of 2.3 1.3 f ( x ) d x , rounded to 10 − 2 . Consider the continuous random variable X which, for a corn plant taken at random from the field under study, associates https://chingmath.fr chapExoCorrec/4175 sacados/4175 chapExoCorrec/4208 sacados/4208 01234512Cf chapExoCorrec/4056 sacados/4056 chapExoCorrec/6407 sacados/6407 1,21,41,61,822,22,4Nombre en milliers24Cf100plantsTaille enm
its height c Determine the probability P 1.9 X 2.2 , rounded to 10 − 2 . d Is there a function g such that for any pair of real num-bers ( a ; b ) satisfying a<b , we have : P a X b = b a g ( x ) d x If so, give an expression for this function? E.6892 At the time of the 2013 population cen-sus, France had 66 million people. For this exercise, we make the assumption (absurd) that the sizes (in m ) of the French are evenly distributed over the interval 1.4 ; 1.9 Consider the random experiment of randomly selecting one person at random from the French population and note X the random variable that returns the height of the selected person. Determine the value of the following probabilities : a P 1.4 X 1.65 b P X 1 ; 6 c P 1.4 X 1.9 d P X =1.6783453 7. Example of a continuous law E.4218 Consider the function f defined by: f ( x ) = 1 40 · x + 1 5 pour x ∈ 0 ; 4 f ( x ) = 0 otherwise 1 Justify that the function f defines a density function on the interval 0 ; 4 . 2 Let X be the random variable defined on 0 ; 4 whose probability distribution has density f . Determine the following probabilities : a P ( X 1) b P ( X 2) c P 1 2 X < 3 E.4220 For the following question, only one of the four statements is correct. Indicate the correct answer; no justification is required. Let f be the function defined on 0 ; 1 by: f ( x )= x + m where m is a real constant. f is a probability density on the interval 0 ; 1 when : a m = − 1 b m = 1 2 c m =e 1 2 d m =e − 1 E.4219 Let m be a real number and f be the function defined on R by: f ( x ) = m · sin x pour x ∈ 0 ; ı f ( x ) = 0 sinon 1 Determine the real m such that f is a probability density on R . 2 Let X be a random variable of which f is a probability density. Determine as a function of x the probability value P ( X x ) 3 Calculate the probability: P ı 4 X 3 · ı 4 . 4 Calculate the probabilities : P ( X 0) ; P ( X 0) . 8. Uniform law E.4216 Say whether the following propo-sition is true or false and justify the answer given : If X is a random variable following the uniform distribution on 0 ; 1 , alors P 0.1 X 0.6 =0.6 E.4217 Of the four proposals presented, only one is correct. Give the correct answer. It is assumed that the waiting time at a service counter, ex-pressed in hours, follows a uniform distribution on the interval 0 ; 1 The probability that a random person’s waiting time is be- tween 15 min and 20 min. a 1 3 b 1 5 c 1 12 d 1 4 E.5466 All staff in a hospital have a home-hospital travel time of at most one hour, and the exact travel time is assumed to be a random variable uniformly distributed over 0 ; 1 . A member of staff at this hospital is interviewed at random. What is the probability that the interviewee has a travel time between 15 min and 20 min? 9. Introduction to continuous laws E.7381 Consider the following random experi-ment in which a dart is thrown at random at a target. For modeling purposes, it is assumed that each time the dart is thrown, it lands in the target. https://chingmath.fr chapExoCorrec/6892 sacados/6892 chapExoCorrec/4218 sacados/4218 chapExoCorrec/4220 sacados/4220 chapExoCorrec/4219 sacados/4219 chapExoCorrec/4216 sacados/4216 Extrait d'Antilles-Guyanes Septembre 2009 chapExoCorrec/4217 sacados/4217 Extrait d'Antilles-Guyanes Juin 2010 chapExoCorrec/5466 sacados/5466 Extrait de Liban Juin 2004 chapExoCorrec/7381 sacados/7381
CibleACibleB24cm6cm 01234512Cf 1 Consider the plastic target A where the plastic darts hit-ting the target come to lodge in one of the holes shown in the figure equiprobably. What is the probability that the dart will lodge in the center circle of the dartboard? 2 Consider the target B made of cork, where the steel-tipped darts can equiprobably lodge anywhere on the target. What is the probability that the shooter will place the dart on the center circle of the dartboard? E.7382 Shown below is the representative curve of the function f defined on the interval 1 ; 5 by the rela-tion : f ( x ) = 32 (3 x + 1) 2 Consider a dart-throwing game based on the shaded surface defined by: the x-axis and the curve C f ; the straight lines with equations x =1 and x =5 . Assuming that on each throw, the random dart falls into this shaded area, we wish to know the probability of the dart reaching the hatched area bounded by the two straight lines x =3 and x =5 . To do this, consider the random variable X which associates with each dart thrown the abscissa of its point of reception on the target. 1 Using a calculator, give the values of the integrals below : 5 1 f ( x ) d x ; 5 3 f ( x ) d x 2 Determine probability: P 3 X 5 10. Continuous law and calculator E.7383 Consider the function f defined by: f ( x ) = 1 40 · x + 1 5 for x ∈ 0 ; 4 f ( x ) = 0 sinon 1 Justify that the function f defines a probability density function on the interval 0 ; 4 . Using a calculator, give the value of the integral: 4 0 f ( x ) d x . 2 Let X be the random variable defined on 0 ; 4 whose probability distribution has density f . Using a calcula- tor, determine the probabilities : a P ( X 1) b P ( X 2) c P 1 2 X < 3 E.7384 Let f be the function defined on 0 ; 1 by: f ( x )= x + 1 2 Justify that the function f is a probability density on the in-terval 0 ; 1 . 1 0 f ( x ) d x will be determined using the calculator. 11. Uniform law E.7395 Say whether the following propo-sition is true or false and justify the answer given : If X is a random variable following the uniform distribution on 0 ; 1 , alors P 0.1 X 0.6 =0.6 E.7396 Of the four proposals presented, only one is correct. Give the correct answer. It is assumed that the waiting time at a service counter, ex-pressed in minutes, follows a uniform distribution on the in-terval 0 ; 60 The probability that a random person’s waiting time is be-tween 15 min and 20 min. a 1 3 b 0.2 c 1 12 d 0.25 https://chingmath.fr CibleACibleB24cm6cm chapExoCorrec/7382 sacados/7382 01234512Cf chapExoCorrec/7383 sacados/7383 chapExoCorrec/7384 sacados/7384 chapExoCorrec/7395 sacados/7395 Extrait bac S 2009 chapExoCorrec/7396 sacados/7396 Extrait bac S 2010
E.7397 Every day, Guy plays an online game with three friends. Paul logs on to the site. The time D (in seconds) it takes to reunite the four players is a random variable that follows a uniform distribution on the interval 20 ; 120 . Determine the probability that all four players are together after 60 seconds. E.7832 Of the answers below, only one is correct. Which one? Justify your answer. A random variable X follows a uniform distribution on the interval 1 ; 9 then : a P 1 < X < 9 = 1 8 b P 5 < X < 9 = 1 2 c P 1 < X < 3 = 3 8 d P 1 < X < 2 = 1 2 12. Uniform law and expectation E.7837 Of the answers given, only one is correct. Which one? Justify your answer. In a ski resort, the waiting time at a given chairlift, expressed in minutes, can be modeled by a random variable X follows a uniform distribution on the interval 0 ; 5 . a L the expectation of this law X is 2 5 b P X > 2 = 3 5 c P X 2 = 3 5 d P X 5 = 0 13. Course - Probability E.4267 Let X be a random variable follow-ing an exponential distribution – where – is a strictly positive real number. Reminders: for all t 0 , we have : P ( X t ) = t 0 – · e − λ · x d x The function R defined on the interval 0 ; + ∞ by: R ( t )= P ( X >t ) is called the reliability function. 1 Prove that for all t 0 , we have : R ( t )=e − λ · t 2 Prove that the variable X follows a lifetime distribution without aging, i.e., for all real numbers s 0 , the con-ditional probability P X >t ( X >t + s ) does not depend on the number t 0 . E.4318 Let X be a random variable follow-ing an exponential law with parameter – ( – strictly positive) , i.e. the probability is expressed by: F ( t ) = P X t = P [0 ; t ] = t 0 – · e − λ · x d x Organized knowledge transfer : Prerequisites: P B ( A )= P ( A ∩ B ) P ( B ) where A and B are two events such that P ( B ) =0 ; P A =1 −P ( A ) where A is an event ; P [ a ; b ] = F ( a ) − F ( b ) where a and b are positive real numbers such that a b . Show that, for any positive real number s , we have : P [ t ; + ∞ [ [ t ; t + s ] = F ( t + s ) − F ( t ) 1 − F ( t ) and that P [ t ; + ∞ [ [ t ; t + s ] is independent of the real number t . E.5507 Let X be a random variable following an exponential distribution with parameter – . The expectation of this random variable has the value : E ( X ) = lim x ↦→ + ∞ x − x t · – e λt d t = 1 – E.5508 Let X be a random variable following an exponential distribution with parameter – . For any positive real t and h , we have equality: P ( X t ) X t + h = P ( X h 14. Uniform and exponential laws https://chingmath.fr chapExoCorrec/7397 sacados/7397 chapExoCorrec/7832 sacados/7832 Extrait Centres Etrangers Juin 2017 chapExoCorrec/7837 sacados/7837 Extrait Antilles-Guyane Juin 2017 chapExoCorrec/4267 sacados/4267 chapExoCorrec/4318 sacados/4318 Extrait d'Asie Juin 2011 chapExoCorrec/5507 sacados/5507 chapExoCorrec/5508 sacados/5508
234I2JO E.3206 The lifetime of a robot, expressed in years until the first breakdown occurs, is a random variable that follows an exponential law of parameter – , with –> 0 . Thus, the probability of a robot falling into a pass before time t is equal to : P ( X t ) = t 0 – e − λx dx 1 Determine – , rounded to the nearest 10 − 1 , so that the probability P ( X > 6) is equal to 0.3 . For the rest of the exercise, we’ll take – =0.2 2 At what time t , to the nearest month, is the probability of a robot breaking down for the first time 0.5 ? 3 Show that the probability of a robot not having broken down in the first two years is e − 0.4 . 4 Knowing that a robot has not had a breakdown in the first two years, what is, to the nearest 10 − 2 , the proba-bility that it will still be in working order after six years? 5 Consider a batch of 10 robots operating independently. Determine the probability that, in this batch, there is at least one robot that has not broken down in the first two years. E.3181 Part A Let X be a continuous random variable that follows an expo-nential distribution with parameter – . Recall that : P ( X a ) = a 0 – · e − λt d t The curve shown below represents the associated density func-tion : 1 Interpret the probability P ( X 1) on the graph. 2 Indicate on the graph where the parameter – can be read directly. Part B We set – =1.5 . 1 Calculate the exact value of P ( X 1) , then its value rounded to 10 − 3 . 2 Calculate P ( X 2) . 3 Deduce the following equality from the previous calcula-tions : P (1 X 2) ≈ 0.173 à 10 − 3 près. 4 Calculate the integral: F ( x )= x 0 1.5 · t · e − 1.5 t d t . Determine the limit when x tends towards + ∞ of F ( x ) ; this gives us the mathematical expectation of the variable X . Part C Note: probabilities will be rounded to 10 − 3 . A machine tool manufactures cylinders. The deviation, in tenths of a millimeter, between the diameter of the cylinders and the machine’s setting value is measured. We assume that this difference follows an exponential distri-bution with parameter – =1.5 . If the deviation is less than 1, the cylinder is accepted. If the deviation is between 1 and 2, the cylinder is rectified so that it can be accepted in 80 % cases. If the deviation is greater than 2, the cylinder is rejected. 1 A cylinder is randomly selected from production. a Show that the probability of him being accepted is equal to 0.916 , rounded to 10 − 3 close. b Knowing that it is accepted, what is the probability that it has undergone rectification? 2 Ten cylinders are independently selected from produc-tion. We assume that the number of cylinders is large enough to treat a draw as a successive draw with replace-ment. https://chingmath.fr chapExoCorrec/3206 sacados/3206 Liban Mai 2006 3 points chapExoCorrec/3181 sacados/3181 Antilles-Guyane Juin 2006 4 points 234I2JO
a What is the probability that all ten cylinders will be accepted? b What is the probability that at least one cylinder will be rejected? E.3164 Parts A and B are indepen-dent Alain is an amateur electronics enthusiast. He buys compo-nents from a store that all appear to be identical, but some of which are defective. The probability that a component sold in the store is defective is estimated to be 0 ; 02 . Part A We assume that the number of components available in the store is large enough that purchasing 50 components is equiv-alent to 50 independent draws with replacement, and we call X the number of defective components purchased. Alain buys 50 components. 1 What is the probability that exactly two of the compo-nents purchased are defective, rounded to the nearest 10 − 1 ? 2 What is the probability that at least one of the compo-nents purchased is defective, rounded to 10 − 2 ? 3 What is the average number of defective components per batch of 50 components purchased? Part B It is assumed that the lifetime T 1 (in hours) of each defective component follows an exponential distribution with parame-ter – 1 =5 × 10 − 4 and that the lifetime T 2 (in hours) of each non-defective component follows an exponential distribution with parameter – 2 =10 − 4 (, refer to the form below) . 1 Calculate the probability that the lifetime of a compo-nent will exceed 1 000 hours a if this component is defective ; b if this component is not defective. Hint: give an approximate value for these probabili-ties to the nearest 10 − 2 . 2 Let T be the lifetime (in hours) of a randomly purchased component. Show that the probability that this component will still be in working order after t hours of operation is : P ( T t ) = 0 ; 02 · e − 5 × 10 − 4 t + 0 ; 98 · e − 10 − 4 t (Remember that the probability that a component sold in the store is defective is equal to 0 ; 02 ) 3 Knowing that the purchased component is still in work-ing order 1 000 hours after installation, what is the prob-ability that this component is defective, rounded to the nearest 10 − 2 ? Form : Exponential law (or lifetime without aging) of parameter – on 0 ; + ∞ For 0 a b , P [ a ; b ] = b a – · e − λx d x For c 0 , P [ c ; + ∞ [ =1 − c 0 – · e − λx d x E.3162 This exercise has two indepen-dent parts. Part I is the demonstration of a course result. The part II is a Q.C.M. Part I : course question Let A and B be two independent events. Show that A and B are independent. Part II For each of the following questions, one and only one of the four propositions is correct. The candidate will indicate on his copy the number of the question and the letter corresponding to the chosen answer. No justification is required. A correct answer earns 1 point. A false answer deducts 0.5 point. The absence of an answer is counted as 0 points. If the total for this part is negative, the mark corresponding to part II is reduced to zero. 1 An urn contains five black balls and three red balls indis-tinguishable to the touch. (out of program 2012) Three balls are simultaneously extracted from the urn. What is the probability of obtaining two black balls and one red ball? a 75 512 b 13 56 c 15 64 d 15 28 2 During a flu epidemic, one-third of a population is vacci-nated. Of those with flu, one in ten is vaccinated. The probability that a person chosen at random from the pop-ulation has the flu is 0.25 What is the probability of a vaccinated individual in this population contracting influenza? a 1 120 b 3 40 c 1 12 d 4 40 3 A player rolls a well-balanced die once. He wins 10 e if the die scores 1. He wins 1 e if the die scores 2 or 4 . He wins nothing in the other cases. Let X be the random variable equal to the player’s win. What is the variance of X ? a 2 b 13 c 16 d 17 4 Waiting time T , in minutes, at a freeway tollbooth be-fore passing through the checkout is a random variable that follows an exponential distribution with parameter – = 1 6 . So for any real t> 0 : P ( T <t ) = t 0 – · e − λx dx (avec – = 1 6 ) where t denotes time expressed in minutes. Knowing that a motorist has already waited 2 minutes, what is the probability (rounded to 10 − 4 ) that his total time is less than 5 minutes? a 0.2819 b 0.3935 c 0.5654 d 0.6065 https://chingmath.fr chapExoCorrec/3164 sacados/3164 chapExoCorrec/3162 sacados/3162
15. Unclassified financial years E.8136 We study certain characteristics of a supermarket in a small town. Part A - Preliminary demonstration Let X be a random variable that follows the exponential dis-tribution with parameter 0.2 . Recall that the expected value of the random variable X , de-noted E ( X ) , is equal to : lim x ↦→ + ∞ x 0 0.2 · t · e − 0.2 · t d t The goal of this part is to demonstrate that E X =5 . 1 Let g be the function defined on the interval 0;+ ∞ by: g ( t )=0.2 · t · e − 0.2 · t . We define the function G on the interval 0 ; + ∞ by: G ( t )= − t − 5 · e − 0.2 · t Verify that G is a primitive of g on the interval 0 ; + ∞ . 2 Deduce that the exact value of E X is 5 . Hint : you can use the following result without proving it: lim x ↦→ + ∞ x · e − 0.2 · x = 0 Part C - Waiting time for payment This supermarket allows customers to choose between using self-service payment terminals or going through a checkout managed by an operator. 1 The waiting time at a self-service terminal, expressed in minutes, is modeled by a random variable that follows the exponential distribution with parameter 0.2 min − 1 . a Give the average waiting time for a customer at an automatic payment terminal b Calculate the probability, rounded to 10 − 3 , that a cus-tomer’s waiting time at an automatic payment termi-nal will exceed 10 minutes. 2 The study commissioned by the manager leads to the following model : among customers who have chosen to use an automatic terminal, 86 % wait less than 10 minutes ; among customers going to the checkout, 63 % wait less than 10 minutes. We choose a random customer in the store and define the following events : B : ˇ the customer pays at an automatic terminal ı ; B : ˇ the customer pays at a cash register with an op-erator ı ; S : ˇ the customer’s waiting time during payment is less than 10 minutes ı Waiting more than ten minutes at a cash register with an operator or at an automatic terminal gives the customer a negative perception of the store. The manager wants more than 75 % of customers to wait less than 10 minutes. What is the minimum proportion of customers who must choose an automatic payment terminal for this objective to be achieved? Part D - Vouchers When paying, scratch cards, either winning or losing, are dis-tributed to customers. The number of cards distributed de-pends on the amount of the purchase. Each customer is enti-tled to one scratch card per 10 e purchase. For example, if the purchase amount is 58.64 e , , the customer receives 5 cards ; if the amount is 124.31 e , , the customer re-ceives 12 cards. The winning cards represent 0.5 % of the total stock of cards. Furthermore, this stock is large enough to assimilate the dis-tribution of a card to a draw with replacement. 1 A customer makes purchases for an amount of 158.02 e . What is the probability, rounded to 10 − 2 , that they will receive at least one winning card? 2 At what purchase amount, rounded to 10 e , , is the prob-ability of receiving at least one winning card greater than 50 % ? E.6247 Let X be a random variable that follows an exponential law of parameter – . Show that, for any positive real t , we have the equality: P ( X t ) X t + h = P X h E.6089 Part A The service life, expressed in years, of a motor to automate a gate manufactured by a company A is a random variable X that follows an exponential law of parameter 0.081 . 1 Determine : P X 3 2 The motor has already been running for 3 years. What is the probability that it will run for another 2 years? Part B The company B produces presence sensors to automatically engage the opening and closing of doors. The lifetime of these sensors, expressed in years, is a random variable Y following an exponential law with parameter – , where – is a strictly positive real. 3 We know that : P Y 2 =0.7 Determine the exact value of the real – . E.4261 The lifetime, expressed in years, of a device is modeled by a random variable X which follows the exponential law with parameter – =0 ; 07 on 0 ; + ∞ . Recall that for any t> 0 , the probability of the event X t is given by: P ( X t ) = t 0 – · e − λ · x d x avec – =0.07 1 Determine the probability that the device will have a lifetime greater than 10 years. 2 Knowing that the device has operated 10 years, deter-mine the probability that it will operate another 10 years. https://chingmath.fr sacados/8136 chapExoCorrec/6247 sacados/6247 chapExoCorrec/6089 sacados/6089 chapExoCorrec/4261 sacados/4261
E.4270 The waiting time T , in minutes, at a freeway tollbooth before passing through the checkout is a random variable that follows an exponential distribution with parameter – = 1 6 . So for any real t> 0 (denoting time expressed in minutes) : P ( T <t ) = t 0 – · e − λ · x d x avec – = 1 6 Knowing that a motorist has already waited 2 minutes, deter-mine the probability, rounded to the nearest 10 − 4 , that his total time will be less than 5 minutes. E.4250 Consider a variable X that follows an exponential distribution of parameter – with –> 0 . Thus, for any positive real t , the probability that the variable X is less than t is given : P ( X t ) = t 0 – · e − λ · t d t Determine – knowing that : P ( X > 5)=0.4 E.4265 The lifetime of electronic compo-nents is being studied. This lifetime is modeled by a random variable X following an exponential law with parameter – , with – a strictly positive real. Knowing that P ( X > 5)=0.325 , determine – . E.4271 A high school physics laboratory has a fleet of identical oscilloscopes. The lifetime in years of an oscilloscope is a random variable noted X that follows the law ˇ lifetime law without vieillissement ı (or exponential law) of parameter 0 ; 125 . All probabilities will be given to the nearest 10 − 3 . 1 The lifetime of an oscilloscope is considered to be inde-pendent of that of other equipment. The laboratory man-ager decides to order 15 oscilloscopes. What is the prob-ability that at least one oscilloscope will have a lifetime of more than 10 years? 2 How many oscilloscopes would the facility have to pur-chase for the probability of at least one oscilloscope op-erating for more than 10 years to be greater than 0.999 ? https://chingmath.fr chapExoCorrec/4270 sacados/4270 chapExoCorrec/4250 sacados/4250 Extrait d'Amerique du Nord Juin 2011 chapExoCorrec/4265 sacados/4265 chapExoCorrec/4271 sacados/4271