Probability distribution of a discrete random variable X P(X = x) E(X) x- (x-)^2*P(x) 0.000008 0.000000 0.000130 0.000130 -8.500000 0.00055122 -7.500000 0.00729561 0.001038 0.002075 -6.500000 0.0438385 0.005188 0.015564 -5.500000 0.15693665 0.018158 0.072632 0.047211 0.236053 0.094421 0.566528 0.148376 1.038635 0.185471 1.483765 0.185471 1.669235 0.148376 1.483765 0.094421 1.038635 0.047211 0.566528 0.018158 0.236053 0.005188 0.072632 0.001038 0.015564 0.000130 0.002075 0.000008 0.000130 8.500000 -4.500000 0.36769867 -3.500000 0.57833099 -2.500000 0.59013367 -1.500000 0.33384705 -0.500000 0.04636765 0.500000 0.04636765 1.500000 0.33384705 2.500000 0.59013367 3.500000 0.57833099 4.500000 0.36769867 5.500000 0.15693665 6.500000 0.0438385 7.500000 0.00729561 8.500000 0.00055122 X 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 a. Find P(X27) 4.25 b. Show your work on this worksheet that the Expected value of X is 8.5. 8.5 c. Show your work on this worksheet that the standard deviation of X is 2.06. 2.06 Recall that the mean and the standard deviation formulas for discrete random variable X are as follows: (@xp(x = x)) o = SD(X) = ((x )P(X = x)) l = E (X) = Cumulative probabilities of X P(X X) 0.000008 0.000137 0.001175 0.006363 0.024521 0.071732 0.166153 0.314529 0.500000 0.685471 0.833847 0.928268 0.975479 0.993637 0.998825 0.999863 0.999992 1.000000 X 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 a. Find P(X 8) b. Find P(5 x11) c. Find P(X=10) Suppose X is a Binomial random variable with the following parameters: n = 308 E(X) = np, o = npq 0.95 0.05 p= q= a. Find P(X 301) b. Find P(X 305) c. Find P(300 X 308) d. Find E(X), the expected value of X. e. Find the standard deviation of X. Suppose Y, the fuel efficiency of cars in mpg, is a normal random variable with the following parameters: = 24.3 0 = 6.53 a. Find P(Y 22) c. Find P(21