Could you please explain your answers thoroughly? Thanks in advance

1.

(10 points) Suppose I give a final project to a course that contains a very large number of students. Suppose that I have two Teaching Assistants to grade these final projects. Suppose the number of errors in grading per project made by the first TA has a Poisson distribution with mean value # and the number of errors in grading per project made by the second TA has a Poisson distribution with mean value /2. If each TA takes an equal share of the final projets to grade then it is reasonable that the function P(X = r) = .5 + .5 r! gives the probability of of a randomly selected final project having r grading errors. (a) Show that this is a legitimate probability mass function. (2 points) (b) What is the expected number of grading errors? (3 points) (c) What is the variance of the number of grading errors? (4 points) (d) How might the given function change if the first TA was able to grade 75% of the final projects? (1 point)8 points) You are an art expert. You decide that you want to test your skills so you have the Gallery director bring out 10 real paintings and 10 forgeries. You then have them randomly select 15 of these paintings for analysis. (a) If we let X be the number of real paintings in the selection, what is the probability mass function of X? (3 points) (b) What is the probability that all paintings of one type are in the selection? (2 points) (c) What is the probability that the number of real paintings selected is within one stan- dard deviation of the expected number of real paintings? (3 points)