MA 20205 Probability and Statistics Assignment No. 8 1. The life of a special type of battery is a random variable with mean 40 hrs and standard deviation 20 hrs. A battery is used until it fails, at which point it is replaced by a new one. Assuming a stockpile of 25 such batteries, whose lives are independent, use the Central Limit Theorem to approximate the probability that over 1100 hrs of use can be obtained. 2. Sylvania’s 40-watt light bulbs will burn a random time ππ before failing. Let ππ have mean ππ and standard deviation 100 hours. If ππ of these bulbs are placed on test till they burn out, resulting in observations ππ1 , … , ππππ , how large should ππ be so that the probability that πποΏ½ differs by ππ by less than 50 hours is at least 0.95? 3. A certain component is critical to the operation of an electrical system and must be replaced immediately upon failure. If the mean lifetime of this type of component is 100 hours and its standard distribution is 30 hours, how many of these components must be in stock so that the probability that the system is in continual operation is for the next 2000 hours is at least 0.95? 4. Let ππ1 , … , ππ9 be i.i.d. ππ(ππ, ππ 2 ). Find ππ(3(πποΏ½ − ππ) > 5.58 ππ) . 5. Let ππ1 , … , ππππ , ππππ+1 be i.i.d. ππ(ππ, ππ 2 ) and let πποΏ½ and ππ 2 denote the sample mean and sample variance based on ππ1 , … , ππππ . Find the distribution of οΏ½ ππ ππ+1 οΏ½ ππππ+1−πποΏ½ ππ οΏ½. 6. Let ππ1 , … , ππππ be a random sample from ππ(ππ1 , ππ 2 ) population and ππ1 , … , ππππ be another independent random sample from ππ(ππ2 , ππ 2 ) population. Let πποΏ½ , πποΏ½ , ππ12 , ππ22 be the sample means and sample variances based on ππ and ππ - samples respectively. Further, let ππππ2 = ππ = (ππ−1)ππ12 +(ππ−1)ππ22 ππ+ππ−2 πΌπΌ(πποΏ½−ππ1 )+π½π½(πποΏ½ −ππ2) ππππ οΏ½οΏ½ πΌπΌ2 π½π½2 + οΏ½ ππ ππ . Determine the distribution of , πΌπΌ ≠ 0, π½π½ ≠ 0. 7. Consider two independent samples- the first of size 10 from a normal population with variance 4 and the second of size 5 from a normal population with variance 2. Compute ππ 2 the ππ οΏ½ππ22 > 3.21οΏ½, where ππ12 andππ22 are the sample variances of the first and second 1 sample respectively. 8. The temperature at which certain thermostat are set to go on is normally distributed with variance ππ 2 . A random sample is to be drawn and the sample variance ππ 2 ππ 2 computed. How many observations are required to ensure that ππ οΏ½ππ2 ≤ 1.8οΏ½ ≥ 0.95? 9. Let ππ1 and ππ2 be independent ππ(0, ππ 2 ) random variables. Find ππ(ππ12 + ππ22 ≤ ππ 2 ). 10. Let ππ1 , … , ππππ be a random sample from ππ(ππ, ππ 2 ) population. For 1 < ππ < ππ, define ππ ππ ππ=ππ+1 ππ=1 ππ 2 ππ = ππ2 = 1 1 1 1 ππ = οΏ½ ππππ , ππ = οΏ½ ππππ , ππ 2 = οΏ½(ππππ − ππ)2 , ππ ππ − ππ ππ − 1 ππ=1 (ππ − ∑ππ ππ−ππ−1 ππ=ππ+1 ππ (ππ−1)ππ 2 +(ππ−ππ−1)ππ 2 )2 ππ . Find the distributions of ππ1 = , ππ3 = ππ2 ππ 2 ππ 2 , ππ4 = √ππ(ππ−ππ) ππ ππ+ππ 2 and ππ5 = , οΏ½(ππ−ππ)(ππ−ππ) ππ . 11. Let ππ1 , ππ2 , … , ππ25 be a random sample from a ππ(2, 4) population and let πποΏ½ and ππ 2 denote the sample mean and the sample variance respectively. Find ππ(2 − 0.3422 ππ < πποΏ½ < 2 + 0.4984 ππ). 12. Let (ππ1 , … , ππ9 ), (ππ1 , ππ2 , … , ππ9 ) and (ππ1 , ππ2 , … , ππ25 ) be independent random samples from ππ(−1, 9), ππ(1, 16) and ππ(0, 9) populations respectively. Find the distribution of ππ = ∑9ππ=1(ππππ +ππππ )2 2 ∑25 ππ=1 ππππ . 13. Let (ππ1 , … , ππ7 be a random sample from a normal population with variance 2 and let ππ1 , ππ2 , … , ππ9 be an independent random sample from a normal population with variance 4. Let ππ12 and ππ22 denote the variances of ππ and ππ samples respectively. Find ππ οΏ½0.12057 < ππ12 ππ22 < 1.7903οΏ½. 14. Let (ππ1 , … , ππ25 ) be independent and identically distributed ππ(3, 5) random variables. Let πποΏ½ and ππ 2 denote the sample mean and the sample variance respectively. Find οΏ½ − 3)2 , ππ = distributions of ππ = 5(ππ 5(πποΏ½−3) ππ and ππ = 25(πποΏ½−3)2 ππ 2 .
