1. (Total: 15 points) (a) (5 points) The lifetime of a light bulb follows an exponential distribution with mean 4 hours. Determine the median lifetime of the light bulb. (b) (5 points) Two independent runners A and B race in a competition. A takes X minutes and B takes Y minutes to finish the race where X ~ N(70,36) and Y ~N(80, 64). What is the probability that A beats B in the race? (c) (5 points) A child is flipping a biased coin 100 times where the probability of showing a head is 0.64. Suppose that the child would be rewarded $2 each time a head is shown, but would be penalized for $1 each time a tail is shown. Determine the expected value and the standard deviation of the child's gain.