Suppose that the random variable X1,...Xn form n Bernoulli trials with parameter p. Determine the conditional probability that X1=1, given that i=1∑n Xi = k  (k=1,...n)

A binomial experiment for the random variable X was performed with 10 trials. The probability of success was found to 0.44. Determine the following probabilities: [Round to four decimal places.] a.) P(X = 4)= b.) P(X 6) =

A diagnostic test for a certain disease is applied to n individuals known to not have the disease. Let X = the number among the n test results that are positive (indicating presence of the disease, so X is the number of false positives) and p = the probability that a disease-free individual's test result is positive (i.e., p is the true proportion of test results from disease-free individuals that are positive). Assume that only Xi available rather than the actual sequence of test results. (a) Derive the maximum likelihood estimator of p. p = If n = 20 and x = 7, what is the estimate? p = (b) Is the estimator of part (a) unbiased? Yes No (c) If n = 20 and x = 7, what is the mle of the probability (1 − p)5 that none of the next five tests done on disease-free individuals are positive? (Round your answer to four decimal places.)

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