Previous Quiz - Q.2 (Binomial within a binomial)Lecture 6 ExamplesMath 370X - Lecture 6 ExamplesFrequently Used Continuous DistributionsSaumil PadhyaOctober 20, 2016Previous Quiz - Q.2 (Binomial within a binomial)2.A study is being conducted in which the health of two independent groups of ten policyholdersis being monitored over a one-year period of time. Individual participants in the study dropout before the end of the study with probability 0.2 (independently of the other participants).What is the probability that at least 9 participants complete the study in one of the twogroups, but not in both groups?(A) 0.096(B) 0.192(C) 0.235(D) 0.376(E) 0.4691Lecture 6 ExamplesExample 1:If X has a continuous uniform distribution on the interval from 0 to 10, the what isP[X +10X> 7]?Example 2:Three individuals are running a one kilometer race. The completion time for eachindividual is a random variable. Xiis the completion time, in minutes, for person i.X1: uniform distribution on interval [2.9, 3.1]X2: uniform distribution on interval [2.7, 3.1]X3: uniform distribution on interval [2.9, 3.3]The three completion times are independent of each other.a. Find the probability that the earliest completion time is less than 3 minutes?b. Find the probability that the latest completion time is less than 3 minutes?Example 3:For Company A there is a 60% chance that no claim is made during the comingyear. If one or more claims are made, the total claim amount is normally distributed with mean10,000 and standard deviation 2,000. For Company B there is a 70% chance that no claim ismade during the coming year. If one or more claims are made, the total claim amount is normallydistributed with mean 9,000 and standard deviation 2,000. Assumer that the total claim amountsof the two companies are independent. What is the probability that, in the coming year, CompanyB’s total claim amount will exceed Company A’s total claim amount?Example 4:A life insurance company has found there is a 3% probability that a randomlyselected application contains an error. Assume applications are mutually independent in thisrespect. An auditor randomly selects 100 applications. Calculate the probability that 95% or lessof the selected applications are error-free.(A) 0.08(B) 0.10(C) 0.13(D) 0.15(E) 0.18Example 5:Losses due to burglary are exponentially distributed with mean 100. The probabilitythat a loss is between 40 and 50 equals the probability that a loss is between 60 and r, with r >60. Calculate r.Example 6:A piece of equipment is being insured againt early failure. The time from purchaseuntil failure of the equipment is exponentially distributed with mean 10 years. The insurance willpay an amount x if the equipment fails during the first year, and it will pay 0.5x if failure occursduring the second or third year. If failure occurs after the first three years, no payment will bemade. At what level must x be set if the expected payment made under this insurance is to be1000?Example 7:The time until the next car accident for a particular driver is exponentially distributedwith a mean of 200 days. Calculate the probability that the driver has no accidents in the next365 days, but then has at least one accident in the 365-day period that follows this initial
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