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- In bacterial counts with a haemacytometer, the number of bacteria per quadrat has a Poisson distribution with probability mass function f(x), where f(x) = θ x e −θ/x! and θ is to be estimated. If there are many bacteria in a quadrat, it is difficult to count them all, and so the only information recorded is that the number of bacteria exceeds a certain limit c, a large positive integer. In a random sample of n quadrats, it was.3. Find the probability mass function of the random variable Y which represents the number of red marbles out of three red balls drawn at random from an urn containing 5 red marbles and 6 white marbles.Suppose an insect lays a very large number of eggs Y, where Y ~ Po(lambda), and suppose that each egg survives with probability p. Assuming that the egg's survival is independent, on the average, how many eggs will survive?
- A park ranger is searching for bears in a region of the park where on average there are 5 bears per square mile. The bears are solitary independent creatures, so it is reasonable to assume that the numbers of bears in disjoint regions are independent unknowns and that the number expected in any region is proportional to the area of the region. The ranger can also assume that in a very tiny region, say a square inch, it is impossible to find more than one bear. What is the probability (two decimal place accuracy) that he finds less than 12 bears in a given 2 square mile region of the park?Suppose that on a particular power grid, 30 wind power generating stations each have output which is uniformly distributed between 24 and 36 MW. a) What is the (approximate) probability that the total power available from these 30 stations would exceed 950 MW? Assume that the power generated by each station is independent of other stations (which, of course, doesn’t make much sense, but taking into account dependence needs to know about the covariance between stations, so we’ll ignor that for now). b) What total power level from these 30 stations would have only a 5% chance of being exceeded, approximately?20% of the workers in a factory work in plant A and 80% work in plant B. The number workers calling in sick per week follow a Poisson probability mass function with parameter λ=3 for plant A, and λ=5 for plant B. What is the probability that next week there will be 2 workers calling in sick?
- You have a collection of 15 phone cords, 4 of which work to charge your current phone and 11 of which are either broken or belong to old phones. You pick 2 cords randomly from the collection. Let X = the number of working cords among those 2. Write out the probability mass function (pmf) of X in table form.The United States Department of Agriculture (USDA) found that the proportion of young adults ages 20–39 who regularly skip eating breakfast is 0.2380.238. Suppose that Lance, a nutritionist, surveys the dietary habits of a random sample of size ?=500n=500 of young adults ages 20–39 in the United States. Apply the central limit theorem to find the probability that the number of individuals, ?,X, in Lance's sample who regularly skip breakfast is greater than 126126. You may find table of critical values helpful. Express the result as a decimal precise to three places. Then, Apply the central limit theorem for the binomial distribution to find the probability that the number of individuals in Lance's sample who regularly skip breakfast is less than 9898. Express the result as a decimal precise to three places.The United States Department of Agriculture (USDA) found that the proportion of young adults ages 20–39 who regularly skip eating breakfast is 0.2380.238. Suppose that Lance, a nutritionist, surveys the dietary habits of a random sample of size ?=500n=500 of young adults ages 20–39 in the United States. Apply the central limit theorem to find the probability that the number of individuals, ?,X, in Lance's sample who regularly skip breakfast is greater than 126126. You may find table of critical values helpful. Express the result as a decimal precise to three places.
- The manufacturer of a fertilizer guarantees that, with the aid of the fertilizer, 70 % of planted seeds will germinate. Suppose the manufacturer is correct. If 7 seeds planted with the fertilizer are randomly selected, what is the probability that at most 5 of them germinate?Carry your intermediate computations to at least four decimal places, and round your answer to two decimal places.1 What is the probability that a SSRW on Z^2 starting at (0, 0) eventually hits the point (100, 100)?