Subject:MathPrice: Bought3
. The ?nal exam in CSE 312 consists of 10 pages, and each student must upload exactly 10 images: one for each of the 10 pages in the exam. Careless students - Shuf?e the 10 images randomly, with each possible ordering equally likely. - Rotate each image randomly (and independently of other images), with each of the 4 rotations equally likely. Only one of these rotations is upright. We say a single image is perfect if it has perfect position (it is in the correct position in the exam), AND has perfect orientation (is upright). On the other hand, careful students make sure all ten of their images are perfect. (It is possible though unlikely that a careless student has all ten perfect pages as well.) (a) What is the expected number of perfect images for a careless student? Hint: The range of this RV is {0, 1,2, . . . ,9, 10}. Give your answer to four decimal places. (b) What is the variance of the number of perfect images for a careless student? Give your answer to four decimal places. (c) Alex thinks that the number of careless students (out of 70) is equally likely to be any integer in {0, 1, 2,. . . ,70} (the remaining students are careful). What is the expected number of perfect images (out of the 700 total images uploaded)? Hint: Be very careful of your expression for this quantity, and remember that a careful student submits 10 perfect images. Give your answer to four decimal places. (d) The deadline has passed, and Alex knows now that exactly 50 students were careless (the remaining 20 students were careful). Compute the probability that at least 220 images are perfect, using the CLT. Give your answer to four decimal places.
Hint:
The positions are not independent because the positioning of a given page is affected by the positioning of other pages. For example, there is some nonzero probability that page 2 is in the second position, but suppose we're given that page 1 is actually in the second position. Then, the probability of page 2 being in the second position is now 0.
We have to use the big variance formula that involves finding the covariance and we can also find the expectation a similar way to that hat problem.
