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Exercise 7.2 If r is the number of heads in 5 tosses of a biased coin for which the probability of heads is 10%, then for any number k between 0 and 5 inclusive, show that Follow P(r = k) = C5,k . (. 1)* . (.9)5-k Inlol holtudruall Exercise 7.2 Find the expected value, variance, and mean of the random variable x with the following probability distribution: X 1 2 3 4 P(x) S .5 .2 .3 0 Exercise 7.3 A fair coin is tossed repeatedly until the first heads appears. Let x be a random variable that counts the number of tosses.B. What are the possible values of r? Hint: In principle, no number of tosses can be excluded. b. What is the probability that r - 17 What is the probability that x - 27 In general, what is the probability that & tosses are required to produce the first heads? Hint: The tosses are independent of each other. C. (Optional) What is the probability that heads never appears? Hint: Intuitively, that probability should be D. Is it? See Example F.5 In Appendix F. Exercise 7.4 A research physician is looking for a person with a very rare genetic disease that afflicts only 1% of the population. She decides to keep on testing randomly chosen people until the first person with the disease is found. What is the probability that the first person with the disease is the sixth person tested? Can you guess (no proof required) the average number of people that need to be tested to get the first person with the disease? twice as likely to land heads as tails is tossed 3 times, her face is rolled. Let x be