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- b) Events A and B are such that P(A) = 3/5, P(B) = 4/15. Find P(A U B) if i) A and B are mutually exclusive events. (3 marks) ii) A and B are independent events. (3 marks)6. Let X1, X2, .., Xn, ... be a sequence of i.i.d. random variables with U[3,7] distribution. Let Mn X1++Xn. Find a number m such that M, m. a.8.A second-stage smog alert has been called in a certain area of Los Angeles County in which there are 90 industrial firms. An inspector will visit 30 randomly selected firms to check for violations of regulations. (a) If 36 of the firms are actually violating at least one regulation, what is th nmf of the number of firms visited by the inspector that are in violation of at least one regulation? O Пx; 30, 0.4) O nb(x; 30, 0.4) O h(x; 30, 36, 90) О Бх;B 30, 0.4) O nb(x; 30, 36, 90) О Бх; 30, 36, 90) (b) If there are 900 firms in the area, of which 360 are in violation, approximate the pmf of part (a) by a simpler pmf. О hix; 30, 360, 900) O nb(x; 30, 0.4) O b(x; 30, 0.4) O nb(x; 30, 360, 900) O h(x; 30, 0.4) О Бix; 30, 360, 900) (c) For X = the number that are in violation among the 30 visited out of 900 firms, compute E(X) and V(X) both for the exact pmf and the approximating pmf in part (b). (Round your answers to two decimal places.) Compute E(X) and V(X) for the exact pmf. E(X) V(X)…
- 2. A number U is selected at random between 0 and 1. Let the events A and B be "U differs from 1/2 by more than 1/4" and "1-U is less than 1/2", respectively. Find the events An B, Ān B, AU B.A fair six-sided die is rolled six times. If an even face numbered k is the outcome onan even roll k for k = 1, . . . , 6, or an odd face numbered k is the outcome on an oddroll k, then we say that a match has occurred. The experiment is called a success if at least one match occurs during the six trials. Otherwise, the experiment is called a failure. (a) Find the outcome space O. (b) Find the probability of failure. (c) Find the probability of success.How can I show the first statement of question 14 and answer the question at the end?
- 5. Suppose 2 = {1,2,3,...} Assume that all subsets are events and that 0 < p<1. Suppose that P({k}) = c(1 − p)k for k € Ω. (a) Determine c. (b) What is P({k}) now that you have determined c? (c) Compute P({k + 1, k + 2,...}) for k = 0, 1, 2, . . ..Prove the Bonferroni inequalities, which state that for any events {A1, A2, ..., An} CF and any integer ke {1, 2,..., n}, k Pr(U²₁_₁A₁) ≤ Σ(-1)²-1 l=1 k Pr(U²1 Aj) ≥ Σ(−1)²-1 l=1 Σ SC1,2,...,n|| St Pr(Njës A;)if k is odd Σ Pr(njes Aj), if k is even SC1,2,...,n||S|=lLet (A,n 2 1) be independent events. Then a)E=1 P(A,) = 0 == P(A i.o.) = 1 b) En=1 P(A,)<=P(A, i.o.) = 0.
- 3 a) The events A and B are such that P(A) =0.3, P(B'| A) =0.8 and P(B| A')=0.4. Find P(AOB) and P(AUB). Hence, state with reason whether A and B are independent events. b) A box contains 5 balls labelled 1, 2, 3, 4 and 5. A player draws a ball from the box randomly. If the number on the ball drawn is 2, 3 or 4, the players score is that number. If the number on the ball is 1 or 5, the player can draw a second ball without replacing the first ball, and his score is the sum of the numbers on the two balls. The events A and B are defined as follows A: The score of a player is 4, 5, 6 or 7. B: A player draws two balls from the box. Find P(A), P(B), P(ANB) and P(AUB).5. A group of n people stand in a line. On the count of three, each of them simultaneously chooses to look either left or right (with equal probability): at one of their neighbors. Let X be the number of pairs of adjacent people that end up facing each other. (For example, if n = 5 and the random facings are "Left, Left, Right, Left, Right" then only the 3rd and 4th people face each other.) (a) Find the expected value E[X]. (b) Find Pr[X=0]. (c) Assuming n is even, find the maximum possible value of X, and the probability that X is equal to that value.3. The following is called the Bonferroni's inequality: For events A and B, we have that: P(ANB) ≥ P(A) + P(B) – 1. - a. Prove the Bonferroni inequality. b. Let A and B be events with probabilities P(A) Show that ≤P(AΜB) ≤ } . 12 = 3 and P(B) = 3.