only do 12

(1) The probability that an "accident prone" policy holder has an accident within a year is 0.6, while the probability that a "non-accident prone" policyholder has an accident within a year is 0.2. Assume that 10% of the policyholders are "accident prone". If a policyholder had an accident, what is the probability that this policyholder is accident prone? (2) If three dice are rolled, and X is the number of twos, find EX?. (3) Let X be the continuous random variable uniformly distributed on the interval [1, 3]. Find the prob- ability density function for Y = X2. (4) Suppose the dice is rolled 180 times. Use the normal approximation to estimate the probability that six occurred 29 times. Express your answer using @, but do not use the normal table. (5) Let S2 = X1 + X2 is the sum of two independent random variables, and each X; is an exponential random variable with A = ,. For $2, find the moment generating function m(t), m'(0) and m"(0). (6) Let $25 = X1 + ... + X25 is the sum of 25 independent random variables, and each X, is Poisson with A = 4. Use the CLT to estimate the probability that $25 > 90. Express your answer using &, use the continuity correction, but do not use the normal table. 7) Suppose that X1 and X2 are independent random variables with EX, = EX2 = 0 and VarX] = VarX2 = 1. Let Y1 = X1 +3X2 and Y2 = 3X1 + X2. Find the correlation coefficient p( Y1, Y2). (8) Suppose that random variables X and Y are uniformly distributed in the region r > 0, y > x, y 0, y > x, y