1. Based on extensive testing, it is determined that the time X in ycars before a major repair is required for a certain washing machine is characterized by the exponential decnsity function with mcan 4. The machine is considered a bargain if it is unlikcly to require a major repair before the sixth year. What is the probability P(3 < X < 6)?
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- Show that if X is not a deterministic random variable, then H(X) is strictly positive. What happens to the probabilities if a random variable is non-deterministic?A city uses three pumps to carry water from a river to a reservoir. Pumps A and B are new, and have a probability of failing of 0.015 on any day. Pump C is older, and has a probability of failure of 0.08 on any day. Pumps A and B operate Monday-Friday. On Saturday, pumps A and C operate while pump B is serviced. On Sunday, pumps B and C operate while pump A is serviced. Answer the following questions, assuming that the pumps operate independently of one another, and independently from day to day. Determine the probability that pump A works on a day that it is in use. Find the probability that pumps A and B both fail on a day they are both in use. Compute the probability that at least one pump fails on any Sunday. Find the probability that pump A works, and C fails on any Saturday. Determine the probability that no pumps fail in a week.An instructor has given a short quiz consisting of two parts. For a randomly selected student, let X = the number of points earned on the first part and Y = the number of points earned on the second part. Suppose that the joint pmf of X and Y is given in the accompanying table. y p(x, y) 0 5 10 15 x 0 0.01 0.06 0.02 0.10 5 0.04 0.16 0.20 0.10 10 0.01 0.15 0.14 0.01 (a) Compute the covariance for X and Y. (Round your answer to two decimal places.)Cov(X, Y) = (b) Compute ρ for X and Y. (Round your answer to two decimal places.)ρ =
- Let (Ω, Pr) be a probability space, and let X and Y be two independent random variables that are positive and have non-zero variance. (a) Prove that X2 and Y are independent. Note that by symmetry, it will also follow that Y 2 and X are independent. (b) Use the result from part (a) to show that the random variables W = X + Y and Z = XY are positively correlated (i.e. Cov(W, Z) > 0).Let X and Y are independent Poisson random variables such that E(X) = E(Y)=2. Let Z=X+Y. Compute P(X=2|Z=3).Of all customers purchasing automatic garage-door openers,75% purchase a chain-driven model. Let X 5 thenumber among the next 15 purchasers who select thechain-driven model.a. What is the pmf of X?b. Compute P(X . 10).c. Compute P(6 # X # 10).d. Compute m and s2.e. If the store currently has in stock 10 chain-drivenmodels and 8 shaft-driven models, what is the probabilitythat the requests of these 15 customers can allbe met from existing stock?
- As a generalization of Example 5.3(figure), consider a test of n circuits such that each circuit is acceptable with probability p, independent of the outcome of any other test. Show that the joint PMF of X, the number of acceptable circuits, and Y, the number of acceptable circuits found before observing the first reject, is PX,Y(x,y) = ((n-y-1)C(x-y))*p^(x)*(1-p)^(n-x) For 0 ≤ y ≤ x < n p^(n) For x=y=n 0 otherwise Hint: For 0 ≤ y ≤ x < n, show that {X = x, Y = y} = A ∩ B ∩ C, where A: The first y tests are acceptable. B: Test y + 1 is a rejection. C: The remaining n − y − 1 tests yield x − y acceptable circuitsLet X1 and X2 be independent random variables; X1 ∼ Poisson(2); X2 ∼ Poisson(3). Compute E(X1|X1 +X2 = 10).4.20. How can I show that X is a Poisson random variable with parameter lambda, then E[Xn] =.... ? And after, using this result to compute E[X3]?
- A k out of n system is one in which there is a group of n components, and the system will function if at least k of the components function. Assume the components function independently of one another. a) In a 3 out of 5 system, each component has probability 0.9 of functioning. What is the probability that the system will function? b) In a 3 out of n system, in which each component has probability 0.9 of functioning, what is the smallest value of n needed so that the probability that the system functions is at least 0.90?Which of the following tests helps in the detection of heteroscedasticity? a.The joint F- test b.The Breusch-Pagan test c.The Chow test d.The Jarque-Berra testOf n1 randomly selected male smokers, X1 smoked filter cigarettes, whereas of n2 randomly selected female smokers, X2 smoked filter cigarettes. Let p1 and p2 denote the probabilities that a randomly selected male and female, respectively, smoke filter cigarettes. (a) Show that (X1/n1) − (X2/n2) is an unbiased estimator for p1 − p2. [Hint: E(Xi) = nipi for i = 1, 2.] E X1 n1 − X2 n2 = 1 n1 E − 1 n2 E = 1 n1 − 1 n2 = p1 − p2 (b) What is the standard error of the estimator in part (a)? (d) If n1 = n2 = 206, x1 = 134, and x2 = 165, use the estimator of part (a) to obtain an estimate of p1 − p2. (Round your answer to three decimal places.)