1. Describe a sample space in rolling two six-side fair dice. Define the events: A= (at...

Transcribed Image Text:

1. Describe a sample space in rolling two six-side fair dice. Define the events: A= (at least one "6" appears), B= (the sum of dots on the dice is divisible by 3). Find AUB, A0B, A-B, B-A, A', B', A'UB', (ANB)'. 2. Formulate the low of total probability and Baye's theorem. Formulate and solve your own example that illustrate these theorems. 3. Let X be a discrete random variable with CDF given in the table Table 1. Cumulative Distribution Function (00, 1) [1, 2) [2, 3) 3,4) 4,00) 0 0.18 0.48 0.86 1.0 H F(x) a. Complete the table for pdf Table 2. Probability distribution function 1 2 IL 3 4 P(₁) b. Sketch CDF and pdf. c. Find parameters m₁ = E(X), m₂ = E(X²), 12 = D²(X), o = D(X). 4. An experiment consists in rolling the six - sided fair die twice. A sample space for this experiment is =w=(i,j): i, j = 1,2,3,4,5,6. The random variable X is defined by rule X(i, j)=i+j, where i = 1,2,3,4,5,6 and j = 1,2,3,4,5,6. a. Determine the probability distribution function of the random variable X. b. Sketch pdf. c. Find parameters m₁ = E(X), m₂ = E(X²), 2= D²(X), o = D(X). 5. Formulate the definition and form of the binomial distribution. Three cards are drown with replacement from a deck of 52 cards. A value of the random variable X denotes the number of spades in this experiment. a) Find probability distribution function of the random variable X. b) Compute the probability of obtaining at list two spades. c) Find the expected value and variation of the random variable X. d) Find CDF of the random variable X. The basis for passing the exam is to resolve these exercises and provide me until 26 May 2017. The rating of the exam depends on well resolving exercises and number of plusses getting during the classes. 1. A box contains 8 numbered balls (1,2,3,4,5,6,7,8). We draw without replacement four balls. Describe an appropriate sample space for this experiment. Find probability that the numbers of all drawn balls are less than 6. 2. Formulate the low of total probability and Baye's theorem. - Three boxes I, II and III contain the balls of two different colors red and white. A box number I contains 2 red and 8 white balls, the box II contains 6 red and 4 white balls and the box III contains 5 red and 5 white balls. We choose a box depending on the result of one die roll. If an event A (1,2,3) occurs we draw ball from a box I, if the event A₂ = (4,5) occurs we draw ball from the box II, if the event As (6) occurs we draw ball from the box II. Let B denote event "the white ball is drawn". a) From the Law of Total Probability find P(B). b) From the Bayes' rule calculate the posterior distribution of events A2, A2, As given event B. 3. The reliability systems which is composed of four components, have the structures given by (21₁, 22, 23, 24)=max(min(₁.ra), min{rs, 74}}. Sketch this reliability structure. We define events A = {"component 1 fails"}, A("component 2 fails"), A = {"component 3 fails"), A = ("component 4 fails"). Probabilities of the components failure are P(A) = P₁=0.02, P(A2)=P2=0.03, P(AS) Pa 0.01 P(A) = P=0.02. Assuming that the events A₁, A2, A3 A4 are mutually independent find reliability of the system (probability that the system is functioning does not fail). 4. Let X be a discrete random variable with CDF F(x)= for z€ (-∞0, 0) ze [0, 1) 0.15 for re [1, 2) 0.35 for ze [2, 3) 0.65 for ze [3, 4) 0.85 for ze [4, 5) 1.00 for ze [5,00) 0 0.05 for a. Find discrete pdf: p(x)=P(X), I, € S. b. Sketch CDF and pdf. c. Find parameters m₁ = E(X), m₂ = E(X²), t2 = D²(X), o = D(X). 5. A soldier shoots at target until a first hitting but no more than 4 times. The probability of hitting the target with a single shot is 0.6. A value of the random variable X denotes the number of shoots until the first hitting but no more shoots than 4. a) Find the pdf of the random variable X. b) Find the expected value and variation of the random variable X. c) Compute the probabilities P(X-4), P(X<3). The basis for passing the exam is to resolve these exercises and provide me until 26 May 2017. The rating of the exam depends on well resolving exercises and number of plusses getting during the classes. 1. From a box containing of 6 distinguishable, numbered balls 1, ..., 6 are chosen consecutively without replacement 4 balls. The digit of the balls create four-digit numbers. Find the probability of the event A: "A four-digit number is less than 4000". 2. Formulate the low of total probability and Baye's theorem. In target shooting there are for the use 6 rifles, 3 of which are equipped with optical sight (viewfinder). The probability of hitting a shot of the rifle with optical sight is 0.95, while the probability of hitting the rifle without of this one is 0.60. Calculate the probability that the target will be hit if the soldier shots once with a randomly choosing rifle. 3. The reliability systems which is composed of four components, have the structures given by (1.72, 73, 74) = max{min (1, 2, 3), (24}}. Sketch this reliability structure. We define events A₁ = {"component 1 fails"}, A2 = { "component 2 fails"), A3 = {"component 3 fails"}, A {"component 4 fails"). Probabilities of the components failure are P(A) = P₁=0.02, P(A₂)=P2=0.03, P(A3)=Ps= 0.01 P(A) = P4 = 0.02. Assuming that the events A₁, A2, A3 A4 are mutually independent find reliability of the system (probability that the system is functioning - does not fail). 4. Let X be a discrete random variable with pdf given in the table Table 1. Probability distribution function 4 म 1 2 3 5 P(2) 0.1 0.2 0.4 0.2 0.1 Complete the table for CDF Т F(x) Table 2. Cumulative Distribution Function. (-∞0, 1) 1.2) [2, 3) 3,4) 4,5) 5,00) b. Sketch CDF and pdf. c. Find parameters m₁ = E(X), m₂ = E(X²), ₂=D²(X), o= = D(X). 5. A soldier shoots at target until a first hitting but no more than 5 times. The probability of hitting the target with a single shot is 0.5. A value of the random variable X denotes the number of shoots until the first hitting but no more shoots than 5. a) Find the pdf of the random variable X. b) Find the expected value and variation of the random variable X. c) Compute the probabilities P(X-4), P(X<3). The basis for passing the exam is to resolve these exercises and provide me until 26 May 2017. The rating of the exam depends on well resolving exercises and number of plusses getting during the classes. 1. Six mariners of different height randomly stand in a line. Find probability that: a) The highest stands at the beginning and at the lowest at the end of row. b) The highest and lowest stand side by side. 2. Formulate the low of total probability and Baye's theorem. Formulate and solve your own example that illustrate these theorems. 3. The reliability systems which is composed of four components, have the structures given by (1.22, 73, 74) = min{max{₁,2), max(23, 24}}. Sketch this reliability structure. We define events A = ("component 1 fails"}, A={ "component 2 fails"), A3 = {"component 3 fails"}, A {"component 4 fails"). Probabilities of the components failure are P(A) = P₁=0.01. P(A₂) - Pa=0.02, P(A) = Ps = 0.02 P(A) = P4 = 0.02. Assuming that the events A1, A2, A3 A4 are mutually independent find reliability of the system (probability that the system is functioning - does not fail). 4. Let X be a discrete random variable with CDF given in the table Table 1. Cumulative Distribution Function (-∞0, 1) (1, 2) 2, 3) 3.4) 4,00) 0.40 0.75 1.0 0 0.12 I F(x) a. Complete the table for pdf Table 2. Probability distribution function 1 3 4 म P(x) 2 b. Sketch CDF and pdf. c. Find parameters m₁ = E(X), m₂ = E(X2), 12 D²(X), o = D(X). 5. A game consists in rolling the a six - sided fair die twice. A sample space for this experiment is ft=w = (i,j): i, j = 1,2,3,4,5,6. The random variable X is defined by rule X(i, j)=max(i, j), where i 1,2,3,4,5,6 and j = 1, 2, 3, 4, 5, 6. a. Determine the probability distribution function of the random variable X. b. Sketch pdf. c. Find parameters m₁ = E(X), m₂ = E(X²), 2= D²(X), o = D(X). The basis for passing the exam is to resolve these exercises and provide me until 26 May 2017. The rating of the exam depends on well resolving exercises and number of plusses getting during the classes. 1 1. Three soldiers shot at a target. Each of them gives their one shot. Let A= (the first soldier hit the target), B= (the second soldier hit the target), C-the third soldier hit the target. Using events A, B, C determine the events: D= (at least one of solders hit the target), E = {exactly one of them hit the target}, F = {exactly two of them hit the target }, G= { no one of them hit the targe. 2. Formulate the low of total probability and Baye's theorem. Formulate and solve your own example that illustrate these theorems. 3. Let X be a discrete random variable with CDF given in the table Table 1. Cumulative Distribution Function (-∞0, 1) (1, 2) 2, 3) 3,4) 4,00) 0.18 0.48 0.86 1.0 0 I F(1) a. Complete the table for pdf Table 2. Probability distribution function 1 2 3 AL 4 P(₁) b. Sketch CDF and pdf. c. Find parameters m₁ = E(X), m₂ = E(X2), 12= D²(X), o = D(X). 4. A game consists in rolling the six-sided fair die twice. A sample space for this experiment is = {w = (i,j): i,j=1,2,3,4,5,6). The random variable X is defined by rule X(i, j) = min(i, j}, where i = 1,2, 3, 4, 5, 6 and j = 1, 2, 3, 4, 5, 6. a. Determine the probability distribution function of the random variable X. b. Sketch pdf. c. Find parameters m₁ = E(X), m₂= E(X²), μ₂ = = D²(X), a = D(X). 5. Formulate definition of the binomial distribution. A soldier shoots at target three times. The probability of hitting the target with a single shot is 0.6. A value of the random variable X denotes the number of hitting the target. a) Find probability distribution function of the random variable X. b) Compute the probability of obtaining at list two hitting the target. c) Find the expected value and variation of the random variable X. 1. A box contains 8 numbered balls (1,2,3, 4, 5, 6, 7, 8). One ball from the box is drawn. Describe an appropriate sample space.For events A={"a drown ball hasnumber less than 3"}, B = {"a selected ball has odd number"}, describe events: An B, AUB, A-B, B-A, A, B,ANB, (AUB). 2. a.Formulate the definition of conditional probability. = b. An experiment consists in rolling a six-sided dice and observing what number appears.Natural sample space is the set 12 (1,2,3,4,5,6). The event A= (1,3,5) occurs if the result of the dice roll will be odd number among numbers belonging to the sample space, the event B = {1,2,3,4} occurs if the result will be number less than 5. Find the conditional probability of an event A, given the event B and the conditional probability of the event B, given A. 3. There are 24 numerated 1,2,...., 24 balls in the box. We randomly draw 5 balls without replacement. Calculate a probability that we will draw 5 balls that have number no greater than 16. 4. Three soldiers independently shoot at target. Everyone gives one shot. The probability of the target being hit by soldier I is 0.6, by soldier II is 0.7, and by soldier III is 0.8. The value of random variable X represents a number of shots that hit the target. Find the probability distribution function and the cumulative distribution function of the random variable X. Sketch these functions. Calculate the expected value of the random variable X. 5. The six-side fair die is rolled until "6" will appear for the first time. value of the random variable X denotes the number of independent trials until the first success. Find probability distribution function of the random variable X. b) Find CDF of the random variable X. c) Compute the probabilities P(X4), P(X> 6). d) Find the expected value and variance of the random variable X.. 1. An experiment consists in tossing a fair coin until the first head appears. Describe an appropriate sample space. Define the events A = {the number of tossing a coin until the first head appears is less than 3). The event B (the number of tossing a a coin until the first head appears is an even number} Find AUB, AnB, A-B, B-A, A', B'. 2. Let X be a discrete random variable with CDF given in the table Table 1. Cumulative Distribution Function (-00, 1) (1, 2) 2, 3) 3,4) 4,00) 0.22 0.54 0.84 1.0 0 I F(x) a. Complete the table for pdf Table 2. Probability distribution function 1 2 3 4 AL p(x₂) b. Sketch CDF and pdf. c. Find parameters m₁ = E(X), m₂ = E(X²), 2=D²(X), o = D(X). 3. An experiment consists in rolling the four - sided fair die twice. A sample space for this experiment is =w = (i,j): ij=1,2,3,4. The random variable X is defined by rule X(i, j) = i +j, where i= 1,2,3,4 and j = 1,2,3,4. a. Determine the probability distribution function of the random variable X. b. Sketch pdf. c. Find parameters m₁ = E(X), m₂= E(X²), #2 = =D² (X), o = D(X). 4. Formulate the definition and rule of the binomial distribution. The six-side fair die is rolled 3 times. A value of the random variable X denotes the number of "6" in three independent trials. Find probability distribution function of the random variable X. b) Find CDF of the random variable X. c) Compute the probabilities P(X2), P(X>1). d) Find the expected value and variance of the random variable X. 5. Two people toss a fair coin 4 times each. Find the probability that they will score the same number of heads. 1. From a box containing of 5 distinguishable, numbered balls 1. ..., 5 are chosen consecutively without replacement 3 balls. The digit of the balls create three-digit numbers. Find the probability of the event A: "A four-digit number is less than 300". 2. Formulate the low of total probability and Baye's theorem. In target shooting there are for the use 10 rifles, 3 of which are equipped with optical sight (viewfinder). The probability of hitting a shot of the rifle with optical sight is 0.9, while the probability of hitting the rifle without of this one is 0.6. Calculate the probability that the target will be hit if the soldier shots once with a randomly choosing rifle. 3. The reliability systems which is composed of four components, have the structures given by (1.22. 23. 24) - min(max(2₁.12.za). (24)). Sketch this reliability structure. We define events A = {"component 1 fails"}, A-{ "component 2 fails"), 4= {"component 3 fails"), A = ("component 4 fails"). Probabilities of the components failure are P(A) = P₁= 0.01, P(A₂)-Pa=0.02, P(A)=Ps=0.02 P(A)=P4=0.02. Assuming that the events A₁, A2, A3 A4 are mutually independent find reliability of the system (probability that the system is functioning - does not fail). 4. Let X be a discrete random variable with pdf given in the table Table 1. Probability distribution function 1 2 IL 3 P) 0.1 0.3 0.4 Complete the table for CDF 4 0.2 Table 2. Cumulative Distribution Function (-00, 1) 1,2) 2, 3) 3,4) 4,00) I F(G) b. Sketch CDF and pdf. e. Find parameters m₁ = E(X), m₂= E(X²), a D²(X), o=. = D(X). 5. A game consists in rolling the a six-sided fair die twice. A sample space for this experiment is f2=w=((,j): i.j-1,2,3,4,5,6. The random variable X is defined by rule X(i, j) maz(i, j), where i=1,2,3,4,5,6andj- 1,2,3,4,5,6. M a. Determine the probability distribution function of the random variable X. b. Sketch pdf. o = D(X). c. Find parameters m₁ = E(X), m₂ = E(X²), #2 D²(X), o =. 1. A box contains 8 numbered balls (1,2, 3, 4, 5, 6, 7, 8). We draw in turn without returning three balls. The digits of the balls form the three digit numbers. Describe an appropriate sample space for this experiment. Find probability that the number is less than 600. 2. Give the definition of independence of two events. A fair coin is tossing three times. Check independence of events A={"a head is obtaind in first toss"}, B={ "a head is obtained in third toss"}. 3. The reliability systems consist of four components have the structures given by (11, 12, 13, 14) = min{max{₁, ₂}, max{23, 24}}. Sketch this reliability str cture. Define events A = {"component 1 fails"}, A2 = { "component 2 fails"), A3= {"component 3 fails"}, A = {"component 4 fails"}. Probabilities of the components failure are = 0.01 P(A₁) P₁=0.02, P(A₂) = P2 = 0.03, P(A) = P3 P(A)= P4= 0.02. Assuming that the events A₁, A2, A3 A4 are mutually independent find reliability of the system (probability that the system is functioning - does not fail). 4. Two soldiers independently shoot at target. Everyone gives one shot. The probability of the target being hit by soldier I is 0.7, by soldier II is 0.7. The value of random variable X represents a number of shots that hit the target. Find the probability distribution function and the cumulative distribution function of the random variable X. Sketch these functions. Calculate the expected value of the random variable of the random X. 5. A soldier shoots at target until a first hitting but no more than 4 times. The probability of hitting the target with a single shot is 0.6. A value of the random variable X denotes the number of shoots until the first hitting but no more shoots than 4. a) Find the pdf of the random variable X. b) Find the expected value and variation of the random variable X. c) Compute the probabilities P(X=4), P(X<3). 1. Describe a sample space in rolling two six-side fair dice. Define the events: A= (at least one "6" appears), B= (the sum of dots on the dice is divisible by 3). Find AUB, A0B, A-B, B-A, A', B', A'UB', (ANB)'. 2. Formulate the low of total probability and Baye's theorem. Formulate and solve your own example that illustrate these theorems. 3. Let X be a discrete random variable with CDF given in the table Table 1. Cumulative Distribution Function (00, 1) [1, 2) [2, 3) 3,4) 4,00) 0 0.18 0.48 0.86 1.0 H F(x) a. Complete the table for pdf Table 2. Probability distribution function 1 2 IL 3 4 P(₁) b. Sketch CDF and pdf. c. Find parameters m₁ = E(X), m₂ = E(X²), 12 = D²(X), o = D(X). 4. An experiment consists in rolling the six - sided fair die twice. A sample space for this experiment is =w=(i,j): i, j = 1,2,3,4,5,6. The random variable X is defined by rule X(i, j)=i+j, where i = 1,2,3,4,5,6 and j = 1,2,3,4,5,6. a. Determine the probability distribution function of the random variable X. b. Sketch pdf. c. Find parameters m₁ = E(X), m₂ = E(X²), 2= D²(X), o = D(X). 5. Formulate the definition and form of the binomial distribution. Three cards are drown with replacement from a deck of 52 cards. A value of the random variable X denotes the number of spades in this experiment. a) Find probability distribution function of the random variable X. b) Compute the probability of obtaining at list two spades. c) Find the expected value and variation of the random variable X. d) Find CDF of the random variable X. The basis for passing the exam is to resolve these exercises and provide me until 26 May 2017. The rating of the exam depends on well resolving exercises and number of plusses getting during the classes. 1. A box contains 8 numbered balls (1,2,3,4,5,6,7,8). We draw without replacement four balls. Describe an appropriate sample space for this experiment. Find probability that the numbers of all drawn balls are less than 6. 2. Formulate the low of total probability and Baye's theorem. - Three boxes I, II and III contain the balls of two different colors red and white. A box number I contains 2 red and 8 white balls, the box II contains 6 red and 4 white balls and the box III contains 5 red and 5 white balls. We choose a box depending on the result of one die roll. If an event A (1,2,3) occurs we draw ball from a box I, if the event A₂ = (4,5) occurs we draw ball from the box II, if the event As (6) occurs we draw ball from the box II. Let B denote event "the white ball is drawn". a) From the Law of Total Probability find P(B). b) From the Bayes' rule calculate the posterior distribution of events A2, A2, As given event B. 3. The reliability systems which is composed of four components, have the structures given by (21₁, 22, 23, 24)=max(min(₁.ra), min{rs, 74}}. Sketch this reliability structure. We define events A = {"component 1 fails"}, A("component 2 fails"), A = {"component 3 fails"), A = ("component 4 fails"). Probabilities of the components failure are P(A) = P₁=0.02, P(A2)=P2=0.03, P(AS) Pa 0.01 P(A) = P=0.02. Assuming that the events A₁, A2, A3 A4 are mutually independent find reliability of the system (probability that the system is functioning does not fail). 4. Let X be a discrete random variable with CDF F(x)= for z€ (-∞0, 0) ze [0, 1) 0.15 for re [1, 2) 0.35 for ze [2, 3) 0.65 for ze [3, 4) 0.85 for ze [4, 5) 1.00 for ze [5,00) 0 0.05 for a. Find discrete pdf: p(x)=P(X), I, € S. b. Sketch CDF and pdf. c. Find parameters m₁ = E(X), m₂ = E(X²), t2 = D²(X), o = D(X). 5. A soldier shoots at target until a first hitting but no more than 4 times. The probability of hitting the target with a single shot is 0.6. A value of the random variable X denotes the number of shoots until the first hitting but no more shoots than 4. a) Find the pdf of the random variable X. b) Find the expected value and variation of the random variable X. c) Compute the probabilities P(X-4), P(X<3). The basis for passing the exam is to resolve these exercises and provide me until 26 May 2017. The rating of the exam depends on well resolving exercises and number of plusses getting during the classes. 1. From a box containing of 6 distinguishable, numbered balls 1, ..., 6 are chosen consecutively without replacement 4 balls. The digit of the balls create four-digit numbers. Find the probability of the event A: "A four-digit number is less than 4000". 2. Formulate the low of total probability and Baye's theorem. In target shooting there are for the use 6 rifles, 3 of which are equipped with optical sight (viewfinder). The probability of hitting a shot of the rifle with optical sight is 0.95, while the probability of hitting the rifle without of this one is 0.60. Calculate the probability that the target will be hit if the soldier shots once with a randomly choosing rifle. 3. The reliability systems which is composed of four components, have the structures given by (1.72, 73, 74) = max{min (1, 2, 3), (24}}. Sketch this reliability structure. We define events A₁ = {"component 1 fails"}, A2 = { "component 2 fails"), A3 = {"component 3 fails"}, A {"component 4 fails"). Probabilities of the components failure are P(A) = P₁=0.02, P(A₂)=P2=0.03, P(A3)=Ps= 0.01 P(A) = P4 = 0.02. Assuming that the events A₁, A2, A3 A4 are mutually independent find reliability of the system (probability that the system is functioning - does not fail). 4. Let X be a discrete random variable with pdf given in the table Table 1. Probability distribution function 4 म 1 2 3 5 P(2) 0.1 0.2 0.4 0.2 0.1 Complete the table for CDF Т F(x) Table 2. Cumulative Distribution Function. (-∞0, 1) 1.2) [2, 3) 3,4) 4,5) 5,00) b. Sketch CDF and pdf. c. Find parameters m₁ = E(X), m₂ = E(X²), ₂=D²(X), o= = D(X). 5. A soldier shoots at target until a first hitting but no more than 5 times. The probability of hitting the target with a single shot is 0.5. A value of the random variable X denotes the number of shoots until the first hitting but no more shoots than 5. a) Find the pdf of the random variable X. b) Find the expected value and variation of the random variable X. c) Compute the probabilities P(X-4), P(X<3). The basis for passing the exam is to resolve these exercises and provide me until 26 May 2017. The rating of the exam depends on well resolving exercises and number of plusses getting during the classes. 1. Six mariners of different height randomly stand in a line. Find probability that: a) The highest stands at the beginning and at the lowest at the end of row. b) The highest and lowest stand side by side. 2. Formulate the low of total probability and Baye's theorem. Formulate and solve your own example that illustrate these theorems. 3. The reliability systems which is composed of four components, have the structures given by (1.22, 73, 74) = min{max{₁,2), max(23, 24}}. Sketch this reliability structure. We define events A = ("component 1 fails"}, A={ "component 2 fails"), A3 = {"component 3 fails"}, A {"component 4 fails"). Probabilities of the components failure are P(A) = P₁=0.01. P(A₂) - Pa=0.02, P(A) = Ps = 0.02 P(A) = P4 = 0.02. Assuming that the events A1, A2, A3 A4 are mutually independent find reliability of the system (probability that the system is functioning - does not fail). 4. Let X be a discrete random variable with CDF given in the table Table 1. Cumulative Distribution Function (-∞0, 1) (1, 2) 2, 3) 3.4) 4,00) 0.40 0.75 1.0 0 0.12 I F(x) a. Complete the table for pdf Table 2. Probability distribution function 1 3 4 म P(x) 2 b. Sketch CDF and pdf. c. Find parameters m₁ = E(X), m₂ = E(X2), 12 D²(X), o = D(X). 5. A game consists in rolling the a six - sided fair die twice. A sample space for this experiment is ft=w = (i,j): i, j = 1,2,3,4,5,6. The random variable X is defined by rule X(i, j)=max(i, j), where i 1,2,3,4,5,6 and j = 1, 2, 3, 4, 5, 6. a. Determine the probability distribution function of the random variable X. b. Sketch pdf. c. Find parameters m₁ = E(X), m₂ = E(X²), 2= D²(X), o = D(X). The basis for passing the exam is to resolve these exercises and provide me until 26 May 2017. The rating of the exam depends on well resolving exercises and number of plusses getting during the classes. 1 1. Three soldiers shot at a target. Each of them gives their one shot. Let A= (the first soldier hit the target), B= (the second soldier hit the target), C-the third soldier hit the target. Using events A, B, C determine the events: D= (at least one of solders hit the target), E = {exactly one of them hit the target}, F = {exactly two of them hit the target }, G= { no one of them hit the targe. 2. Formulate the low of total probability and Baye's theorem. Formulate and solve your own example that illustrate these theorems. 3. Let X be a discrete random variable with CDF given in the table Table 1. Cumulative Distribution Function (-∞0, 1) (1, 2) 2, 3) 3,4) 4,00) 0.18 0.48 0.86 1.0 0 I F(1) a. Complete the table for pdf Table 2. Probability distribution function 1 2 3 AL 4 P(₁) b. Sketch CDF and pdf. c. Find parameters m₁ = E(X), m₂ = E(X2), 12= D²(X), o = D(X). 4. A game consists in rolling the six-sided fair die twice. A sample space for this experiment is = {w = (i,j): i,j=1,2,3,4,5,6). The random variable X is defined by rule X(i, j) = min(i, j}, where i = 1,2, 3, 4, 5, 6 and j = 1, 2, 3, 4, 5, 6. a. Determine the probability distribution function of the random variable X. b. Sketch pdf. c. Find parameters m₁ = E(X), m₂= E(X²), μ₂ = = D²(X), a = D(X). 5. Formulate definition of the binomial distribution. A soldier shoots at target three times. The probability of hitting the target with a single shot is 0.6. A value of the random variable X denotes the number of hitting the target. a) Find probability distribution function of the random variable X. b) Compute the probability of obtaining at list two hitting the target. c) Find the expected value and variation of the random variable X. 1. A box contains 8 numbered balls (1,2,3, 4, 5, 6, 7, 8). One ball from the box is drawn. Describe an appropriate sample space.For events A={"a drown ball hasnumber less than 3"}, B = {"a selected ball has odd number"}, describe events: An B, AUB, A-B, B-A, A, B,ANB, (AUB). 2. a.Formulate the definition of conditional probability. = b. An experiment consists in rolling a six-sided dice and observing what number appears.Natural sample space is the set 12 (1,2,3,4,5,6). The event A= (1,3,5) occurs if the result of the dice roll will be odd number among numbers belonging to the sample space, the event B = {1,2,3,4} occurs if the result will be number less than 5. Find the conditional probability of an event A, given the event B and the conditional probability of the event B, given A. 3. There are 24 numerated 1,2,...., 24 balls in the box. We randomly draw 5 balls without replacement. Calculate a probability that we will draw 5 balls that have number no greater than 16. 4. Three soldiers independently shoot at target. Everyone gives one shot. The probability of the target being hit by soldier I is 0.6, by soldier II is 0.7, and by soldier III is 0.8. The value of random variable X represents a number of shots that hit the target. Find the probability distribution function and the cumulative distribution function of the random variable X. Sketch these functions. Calculate the expected value of the random variable X. 5. The six-side fair die is rolled until "6" will appear for the first time. value of the random variable X denotes the number of independent trials until the first success. Find probability distribution function of the random variable X. b) Find CDF of the random variable X. c) Compute the probabilities P(X4), P(X> 6). d) Find the expected value and variance of the random variable X.. 1. An experiment consists in tossing a fair coin until the first head appears. Describe an appropriate sample space. Define the events A = {the number of tossing a coin until the first head appears is less than 3). The event B (the number of tossing a a coin until the first head appears is an even number} Find AUB, AnB, A-B, B-A, A', B'. 2. Let X be a discrete random variable with CDF given in the table Table 1. Cumulative Distribution Function (-00, 1) (1, 2) 2, 3) 3,4) 4,00) 0.22 0.54 0.84 1.0 0 I F(x) a. Complete the table for pdf Table 2. Probability distribution function 1 2 3 4 AL p(x₂) b. Sketch CDF and pdf. c. Find parameters m₁ = E(X), m₂ = E(X²), 2=D²(X), o = D(X). 3. An experiment consists in rolling the four - sided fair die twice. A sample space for this experiment is =w = (i,j): ij=1,2,3,4. The random variable X is defined by rule X(i, j) = i +j, where i= 1,2,3,4 and j = 1,2,3,4. a. Determine the probability distribution function of the random variable X. b. Sketch pdf. c. Find parameters m₁ = E(X), m₂= E(X²), #2 = =D² (X), o = D(X). 4. Formulate the definition and rule of the binomial distribution. The six-side fair die is rolled 3 times. A value of the random variable X denotes the number of "6" in three independent trials. Find probability distribution function of the random variable X. b) Find CDF of the random variable X. c) Compute the probabilities P(X2), P(X>1). d) Find the expected value and variance of the random variable X. 5. Two people toss a fair coin 4 times each. Find the probability that they will score the same number of heads. 1. From a box containing of 5 distinguishable, numbered balls 1. ..., 5 are chosen consecutively without replacement 3 balls. The digit of the balls create three-digit numbers. Find the probability of the event A: "A four-digit number is less than 300". 2. Formulate the low of total probability and Baye's theorem. In target shooting there are for the use 10 rifles, 3 of which are equipped with optical sight (viewfinder). The probability of hitting a shot of the rifle with optical sight is 0.9, while the probability of hitting the rifle without of this one is 0.6. Calculate the probability that the target will be hit if the soldier shots once with a randomly choosing rifle. 3. The reliability systems which is composed of four components, have the structures given by (1.22. 23. 24) - min(max(2₁.12.za). (24)). Sketch this reliability structure. We define events A = {"component 1 fails"}, A-{ "component 2 fails"), 4= {"component 3 fails"), A = ("component 4 fails"). Probabilities of the components failure are P(A) = P₁= 0.01, P(A₂)-Pa=0.02, P(A)=Ps=0.02 P(A)=P4=0.02. Assuming that the events A₁, A2, A3 A4 are mutually independent find reliability of the system (probability that the system is functioning - does not fail). 4. Let X be a discrete random variable with pdf given in the table Table 1. Probability distribution function 1 2 IL 3 P) 0.1 0.3 0.4 Complete the table for CDF 4 0.2 Table 2. Cumulative Distribution Function (-00, 1) 1,2) 2, 3) 3,4) 4,00) I F(G) b. Sketch CDF and pdf. e. Find parameters m₁ = E(X), m₂= E(X²), a D²(X), o=. = D(X). 5. A game consists in rolling the a six-sided fair die twice. A sample space for this experiment is f2=w=((,j): i.j-1,2,3,4,5,6. The random variable X is defined by rule X(i, j) maz(i, j), where i=1,2,3,4,5,6andj- 1,2,3,4,5,6. M a. Determine the probability distribution function of the random variable X. b. Sketch pdf. o = D(X). c. Find parameters m₁ = E(X), m₂ = E(X²), #2 D²(X), o =. 1. A box contains 8 numbered balls (1,2, 3, 4, 5, 6, 7, 8). We draw in turn without returning three balls. The digits of the balls form the three digit numbers. Describe an appropriate sample space for this experiment. Find probability that the number is less than 600. 2. Give the definition of independence of two events. A fair coin is tossing three times. Check independence of events A={"a head is obtaind in first toss"}, B={ "a head is obtained in third toss"}. 3. The reliability systems consist of four components have the structures given by (11, 12, 13, 14) = min{max{₁, ₂}, max{23, 24}}. Sketch this reliability str cture. Define events A = {"component 1 fails"}, A2 = { "component 2 fails"), A3= {"component 3 fails"}, A = {"component 4 fails"}. Probabilities of the components failure are = 0.01 P(A₁) P₁=0.02, P(A₂) = P2 = 0.03, P(A) = P3 P(A)= P4= 0.02. Assuming that the events A₁, A2, A3 A4 are mutually independent find reliability of the system (probability that the system is functioning - does not fail). 4. Two soldiers independently shoot at target. Everyone gives one shot. The probability of the target being hit by soldier I is 0.7, by soldier II is 0.7. The value of random variable X represents a number of shots that hit the target. Find the probability distribution function and the cumulative distribution function of the random variable X. Sketch these functions. Calculate the expected value of the random variable of the random X. 5. A soldier shoots at target until a first hitting but no more than 4 times. The probability of hitting the target with a single shot is 0.6. A value of the random variable X denotes the number of shoots until the first hitting but no more shoots than 4. a) Find the pdf of the random variable X. b) Find the expected value and variation of the random variable X. c) Compute the probabilities P(X=4), P(X<3).