1. Determine the value c so that each of the following functions can serve as a probability distribution of the discrete random variable X: a) f(x) = c(x2 + 4), x = 0,1, 2,3 b ) f ( x ) = c ( ) ( x ), x = 0, 1, 2 2. Let W be a random variable giving the number of heads minus the number of tails in three tosses of a coin. List the elements of the sample space S for the three tosses of the coin and to each sample point assign a value w of W. Find the probability distribution of the random variable W, assuming that the coin is biased so that a head is twice as likely to occur as a tail. 3. Based on extensive testing, it is determined by the manufacturer of a washing machine that the time Y (in years) before a major repair is required is characterized by the probability density function f(y) = 0.25 e-y/4 y 20 i o, elsewhere (a) Critics would certainly consider the product a bargain if it is unlikely to require a major repair before the fourth year. Comment on this by determining P(Y > 4). (b) What is the probability that a major repair occurs in the first year? 4. The life span in hours of an electrical component is a random variable with cumulative distribution function: F (x ) = 1 -e-x/50, x > 0 elsewhere (a) Determine its probability density function. (b) Determine the probability that the life of such a component will exceed 70 hours