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() b. Sketch CDF and pdf. c. Find parameters m = E(X), m = E(X), 12 = D(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 m = E(X), m = E(X), 2= D(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 A = (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 (21, 22, 23, 24)=max(min(.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) = P=0.02, P(A2)=P2=0.03, P(AS) Pa 0.01 P(A) = P=0.02. Assuming that the events A, 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 m = E(X), m = E(X), t2 = D(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