AMS 310 Fall 2023 HW 2

Standard Deviation = sqrt(n*p*q) = 2.766 5. Let X denote the number of bombs hit per minute in an area of 1 square mile on a Particular day during a war. Suppose X has a Poisson distribution with  = 4. (Lambda = V) a) Find the probability that two bombs hit that area in a minute. P(x) = e -V V x /x! = e -4 4 x /x! (x=0,1,2…..) P(x=2)=e -4 4 2 /2! = 0.1465 b) Find the probability that at most two bombs hit that area in a minute. P(x<=2) = P(x=0)+P(x=1)+P(x=2) = e -4 4 0 /0! + e -4 4 1 /1! + e -4 4 2 /2! = 0.01832+0.07326+0.1465 = 0.238 c) Find the expected number of bomb hits per minute in that area. E(x) = V = 4 6. Two percent of certain model of cars have defective mufflers. Suppose 400 cars of this model are ready to ship. a) Find an approximate probability that at least 5 cars in the shipment have defective mufflers. V=np = 400*2/100 = 8 P(x>=5) = 1-[P(x=0)+P(x=1)+P(x=2)+P(x=3)+P(x=4)] = 1-[e -8 8 0 /0! + e -8 8 1 /1! + e -8 8 2 /2! + e -8 8 3 /3! + e -8 8 4 /4!] = 1-[0.0003355+0.00268+0.01073+0.028626+0.05725] = 0.9003785 b) Find an approximate probability that between 3 and 6 cars (inclusive) in the shipment have defective mufflers. P(3<=x<=6) = P(x=3)+P(x=4)+P(x=5)+P(x=6) = 0.028626+0.05725+0.0916+0.12214 = 0.2996 7. Number of passengers who arrive at the platform in an Amtrack train station for the 2 pm train on a Saturday is a normal random variable with a mean of 180 and a standard deviation of 20. a) (5 pts) With what probability can we assert that there will be between 120 and 240 passengers? Using Chebyshev’s Theorem Upper bound for k = 240 = U+k*P = 240 = 180+k*20

? ( ? ≥ 5) = 1 − ? ( ? ≤ 4) = 1 − 0.58 = 0.42 11. Computers from a particular company are found to last on average for three years without any hardware malfunction, with a standard deviation of two months. At least what percent of the computers last between 31 months and 41 months? Use Chebyshev’s Theorem 41=36+k(2) 5=2k = K=2.5 1-1/k^2 = 1-1/6.25 =0.84 84% 12. Bus route #25 takes a mean time of 50 minutes with a standard deviation of 2 minutes. A promotional poster for this bus system states that “95% of the time bus route #25 lasts from ____ to _____ minutes.” What numbers would you fill in the blanks with? 1-0.95 = 1/k 2 k = 4.47 4.47*2 = 8.94 minutes So, Bus route #25 takes a mean time of 50 minutes with a standard deviation of 2 minutes. A promotional poster for this bus system states that “95% of the time bus route #25 lasts from 41.06 to 58.94 minutes.