Only do(7)(8)

(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 d, 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 S2, 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 X, 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