The lifespan X (years) of a species of pig has pdf f (x) = 3x2e−x^3 , 0 < x < ∞. What is the probability that a pig will live at least 1 year? Given that it lives 1 year, what is the conditional probability that it will live for at least another year?
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f (x) = 3x2e−x^3 , 0 < x < ∞.
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- The life (in years) of a certain type of electrical switch has an exponential distribution with an average lifetime of 2 years. For a single system, what is the probability we can keep the switching mechanism operable for 10 years if we assume we will have to replace a switch due to failures three times during the 10-year period?A study of the isopluvial maps revealed that at Najaf Al-Ashraf a maximum rainfall depth of 175 mm in 18 hr has a return period of 100 years. The probability of an 18 hr rainfall equal to or greater than 175 mm occurring at Najaf Al-Ashraf at least once in 50 years isA certain type of storage battery lasts, on average, 3.0 years with a standarddeviation of 0.5 year. Assuming that battery life is normally distributed: a) Write the equations to be used in formulating the solution to theproblem. b) Find the probability that the battery lasts more than 2.5 years.
- Two drunk start out together at the origin, each having the equal probability of making a step to the left or right along the x-axis. Assume that men make their steps simultaneously. Let N be the total number of steps taken by each man. Assume that a particular instant, they are separated by some distance. In this situation, in a given particular step. If they move towards each other, the separation between them decreases. If the move away from each other, the separation between them increases If they move in the same direction, the separation remains unaltered Then calculate (a) The probability of decreasing their separation (b) The probability of increasing their separation (c) The probability of no separation between them Out of N steps, n1 steps belong to decreasing the separation of them; n2 steps belong to increasing the separation of them and n3 steps belong to no separation. Then, calculate the following. (d) The probability of taking the steps n1, n2 and n3 (e) The…Two drunk start out together at the origin, each having the equal probability of making a step to the left or right along the x-axis. Assume that men make their steps simultaneously. Let N be the total number of steps taken by each man. Assume that a particular instant, they are separated by some distance. In this situation, in a given particular step. If they move towards each other, the separation between them decreases. If the move away from each other, the separation between them increases If they move in the same direction, the separation remains unaltered Out of N steps, n1 steps belong to decreasing the separation of them; n2 steps belong to increasing the separation of them and n3 steps belong to no separation. Then, calculate the following. (a) The probability of taking the steps n1, n2 and n3 (b) The probability that they meet again at N stepsSuppose that there are two identically-looking batteries in a box. The first one should last for a time that is exponentially distributed with parameter λ1 = 1 (in months). The second one should last for a time that is exponentially distributed with parameter λ2 = 2. A battery has been picked randomly and is still working after two months. Given this information, what is the probability that the first battery was picked?
- At the end of summer, the total weight of seeds accumulated by a nest of seed-gathering ants will vary from nest to nest. If the total weight of seeds accumulated by a nest is exponentially distributed with parameter λ = 1/5, (a) What is the probability that the total combined weight of the seeds gathered by 100 nests will be larger than 4 95 pounds by the end of next summer? (b) What is the probability that the average weight of the seeds gathered by the 100 nests is larger than 5.3 ? (c) What assumptions are you making to answer parts (a) and (b)? Do you think those assumptions make sense in the context of this problem? Explain. (d) Tell us about the possibility that the total combined weight of the seeds gathered by 2 nests follows a normal distribution. Justify your answer in the specific context of this problem (ants, nests, seed-gathering). That is, do not just make generic statements that you think apply to every context in the world.A certain type of storage battery lasts, on average, 3.0 years with a standarddeviation of 0.5 year. Assuming that battery life is normally distributed: Set up the required equations in order to formulate your solution to theproblem. Find the probability that the battery lasts more than 3.5 years.As soon as possible! Let X and Y be continuous random variables with joint PDF
- For a given electronic component, it has been shown that it is necessary to restart once every 250 hours. Let X be the number restarts within 1000 hours. (f) Given that the electronic component has been operating continuously for 200 hours, what is the probability that it will operate for another 400 hours?A private pilot wishes to insure his airplane for $200,000. The insurance company estimates that atotal loss will occur with probability 0.002, a 50% loss with probability 0.01, and a 25% loss withprobability 0.1. Ignoring all other partial losses, what premium should the insurance company chargeeach year to realize an average profit of $500?Workers at a large toxic cleanup project are concerned that their white blood cell counts may have been reduced. Let x be a random variable that represents white blood cell count per cubic millimeter of whole blood in a healthy adult. Then μ = 7500 andσ ≈ 1750.† A random sample ofn = 70 workersfrom the toxic cleanup site were given a blood test that showedx = 6920.What is the probability that, for healthy adults,xwill be this low or lower?(a) How does the central limit theorem apply? Explain. The central limit theorem describes the distribution of x as normal with mean μ x = 7500 and σ x≈1750.0.The central limit theorem describes the distribution of x as normal with mean μ x = 7500 and σ x≈209.2. The central limit theorem does not apply because the sample size is too small.The central limit theorem describes the distribution of x as normal with mean μ x = 7500 and σ x≈25.0. (b) ComputeP(x ≤ 6920).(Round your answer to four decimal places.) P(x ≤ 6920)= (c) Based on your answer to…