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# 1) Let X be a random variable with probability mass function Px(n) = c (1/n!) for n = 0,1,2,. and Px (n) = 0 otherwise. a) Compute c. (use expansion...

1) Let X be a random variable with probability mass function Px(n) = c (1/n!) for n = 0,1,2,... and Px (n) = 0 otherwise. a) Compute c. (use expansion c^x = summation (x^k)/k!b) Compute the probability that X is even c) Compute the expectation of X. 2) Suppose we want to test a large number of blood samples in order to see if they contain a certain antibody. To reduce the amount of work, one proceeds as follows. We divide the samples into groups of size k, and these k samples are put together. The resulting mixtures are tested. If the test of a mixture is negative, no further action is required. If it is positive, then the k original samples are individually tested after all, so that in such case, a total of k+1 tests needs to be performed. The samples contain the antibody with probability p, independently of each other.a) what is the probability that a mixture of k samples contains the antibody?b) Let S be the total number of test that needs to be performed when the original number of samples is n=mk. Compute E(s) and var(S) 3) Let X and Y be independent random variables with the same distribution, taking values 0 and 1 with equal probability. Show that E((X+Y))(|X-Y|) = E(X+Y)E(|X-Y|)but that X+Y and |X-Y| are not independent