Form 10f Word Format: Fill & Download for Free

This is a simple application of the Central limit theorem.The output of your Monte Carlo algorithm is a random variable. Its unknown mean [math]\mu[/math] is the value you want to compute. This random variable does also have some unknown variance [math]\sigma^2[/math]. In practice, both the mean and the variance are finite.CLT tells you that if you take the average of [math]n[/math] iterations, you will (roughly) get a normally distributed random variable with the same mean and with variance [math]\sigma^2/n[/math]. In other words, the standard deviation of the new random variable is the old standard deviation divided by [math]\sqrt{n}[/math]. That is the rate of convergence you’ll see.If you don’t know the variance of the original random variable, this is as far as you can get: making four times more samples should give you twice the precision, i.e., one more correct bit.If you do know the original variance, you can also compute some confidence intervals based on the fact that almost all values of a normal distribution lie within a few stdevs around the mean.Here’s a simple example with a Monte Carlo calculation of [math]\pi[/math].from math import sqrt, pi from random import uniform  def single():  x, y = uniform(-1,1), uniform(-1,1)  return 4*( x**2 + y**2 <= 1 )  def average(n):  return sum( single() for _ in range(n) ) / n  sigma_single = (4-pi)**2 * (pi/4) + pi**2 * (1-pi/4)  for k in range(25):  n = 2**k  estimate = average(n)  error = abs(estimate - pi)  sigma_avg = sigma_single / sqrt(n)  print( '{:2} {:.10f} {:.10f} {:.10f}'  .format( k, estimate, error, sigma_avg ) ) Each line printed by the sample code contains:[math]k[/math], the binary log of the sample sizethe estimated value of [math]\pi[/math] obtained by taking the average of [math]n[/math] samplesthe absolute error between the estimate and the actual value of [math]\pi[/math]the standard deviation of the average — if you run the code multiple times with different random seeds, you should almost always observe an absolute error of no more than 4 times this.Example output: 0 4.0000000000 0.8584073464 2.6967662133  1 2.0000000000 1.1415926536 1.9069016767  2 4.0000000000 0.8584073464 1.3483831066  3 3.0000000000 0.1415926536 0.9534508383  4 3.0000000000 0.1415926536 0.6741915533  5 3.6250000000 0.4834073464 0.4767254192  6 2.8750000000 0.2665926536 0.3370957767  7 3.1562500000 0.0146573464 0.2383627096  8 3.1250000000 0.0165926536 0.1685478883  9 3.0859375000 0.0556551536 0.1191813548 10 3.1640625000 0.0224698464 0.0842739442 11 3.1425781250 0.0009854714 0.0595906774 12 3.1494140625 0.0078214089 0.0421369721 13 3.1728515625 0.0312589089 0.0297953387 14 3.1335449219 0.0080477317 0.0210684860 15 3.1440429688 0.0024503152 0.0148976693 16 3.1328125000 0.0087801536 0.0105342430 17 3.1415710449 0.0000216087 0.0074488347 18 3.1460266113 0.0044339577 0.0052671215 19 3.1413269043 0.0002657493 0.0037244173 20 3.1403465271 0.0012461265 0.0026335608 21 3.1433010101 0.0017083565 0.0018622087 22 3.1416063309 0.0000136773 0.0013167804 23 3.1409277916 0.0006648620 0.0009311043 24 3.1415805817 0.0000120719 0.0006583902 

{If the formatting is messed up gain then that is not Quora, not me, and the edit does not work.] There is no place btwn Heaven and Hell in Scripture. In Scripture all believers will go to be with the Lord at death or at His return. . (Lk. 23:43 [cf. 2Cor. 12:4; Rv. 2:7]; Phil 1:23; 2Cor. 5:8 ["we"]; 1Cor. 15:51ff'; 1Thess. 4:17) Note in the latter case all believers were assured that if the Lord returned, which they expected in their lifetime, so would they "ever be with the Lord." (1Thes. 4:17) though they were still undergoing growth in grace, as was Paul, who expressed he would go to be with the Lord at death, yet was not already perfect. (Phil. 3:10f)The only suffering for believers that is manifestly taught as after this life is that of the judgment seat of Christ, which event does not occur at death, but awaits the Lord's return, (1 Corinthians 4:5; 2 Timothy. 4:1,8; Revelation 11:18; Matthew 25:31-46; 1 Peter 1:7; 5:4) and is the suffering of the loss of rewards (and the Lord's displeasure) due to the manner of material one built the church with, which one is saved despite the loss of such, not because of. (1 Corinthians 3:8ff)And being "ever with the Lord" they await the resurrection/rapture of their bodies (the first resurrection - the resurrection of life:" Jn. 5:29a; Rev. 20:6,7,14) when they will return with the LORD from Heaven to fight the battle of Armageddon, and execute judgment, and to reign with with Him for the 1,000 year millennium. (Jude 1:14-15; cf. Mal. 3:18; Mt 16:27; 19:28; Mt 24:30,31; 25:31; Lk. 22:20; 2Ths 1:7,8; Rev 1:7; . 2:27; 5:10; 19:5-20; 20:4-6 ) And then after the Final Judgment, will enter the Holy City of the New Jerusalem from Heaven to be with the Lord for Eternity. (Rev. 21,22) Tanks and glory be to God.In contrast, the Lost, those who rejected Christ - which is even by disobeying the innate sense of morality one originally has from God (Romans 2) which would lead one to the Christ of Scripture - go to Hades/Hell at death, which is a general suffering, where they await the Final Judgment, the Great White Throne judgment, where they are sentenced according to their sins and accountability, the degree of guilt.And it came to pass, that the beggar died, and was carried by the angels into Abraham's bosom: the rich man also died, and was buried; And in hell he lift up his eyes, being in torments, and seeth Abraham afar off, and Lazarus in his bosom. And he cried and said, Father Abraham, have mercy on me, and send Lazarus, that he may dip the tip of his finger in water, and cool my tongue; for I am tormented in this flame. (Luke 16:22-24) Then he said, I pray thee therefore, father, that thou wouldest send him to my father's house: For I have five brethren; that he may testify unto them, lest they also come into this place of torment. (Luke 16:27-28) Then shall he say also unto them on the left hand, Depart from me, ye cursed, into everlasting fire, prepared for the devil and his angels:(Matthew 8:11-12; Luke 3:17; Luke 13:28; Matthew 13:41-42; Matthew 13:49-50; Matthew 22:13; Matthew 24:50-51; Matthew 25:30-32; Matthew 25:31-32; Mark 9:43-48; Revelation 14:10-11;Revelation 19:20;In Roman (the EOS differ: http://peacebyjesus.witnesstoday.org/EOvs.Rome.html)" belief in a place called Purgatory developed due to their false belief that there are usally sins that must be atoned for after death and that beliefs must actually become good enough to be with God via purifying punishments.The Catholic Encyclopedia states that St. Augustine "describes two conditions of men; "some there are who have departed this life, not so bad as to be deemed unworthy of mercy, nor so good as to be entitled to immediate happiness" etc.And thus by the close of the fourth century was taught "a place of purgation..from which when purified they "were admitted unto the Holy Mount of the Lord". For " they were "not so good as to be entitled to eternal happiness". - CATHOLIC ENCYCLOPEDIA: PurgatoryLikewise noted Catholic professor Peter Kreeft states,"...we will go to Purgatory first, and then to Heaven after we are purged of all selfishness and bad habits and character faults." Peter Kreeft, Because God Is Real: Sixteen Questions, One Answer, p. 224In contrast to this salvation by effectual faith, is salvation by grace thru works, as in Roman Catholicism, in which by grace one is actually made good enough to be with God via the act of baptism, even without the required wholehearted repentant faith. (Acts 8:38; 8:36,37)But which progressive practical sanctification is not the cause of the sinner's justification and acceptance in Christ, but testifies to such being a believer, evidencing "things which accompany salvation," (Hebrews 6:9) and fit to be rewarded. (Revelation 3:4) For this faith, as manifested in said obedience, God will recompense (Hebrews 10:35) under grace, even though it is God who motivates and enables all obedience, (Philippians 1:12,13) while the only thing we can and must take credit for it our disobedience.If ye then be risen with Christ, seek those things which are above, where Christ sitteth on the right hand of God. Set your affection on things above, not on things on the earth. For ye are dead, and your life is hid with Christ in God. When Christ, who is our life, shall appear, then shall ye also appear with him in glory. (Colossians 3:1-4)If we live in the Spirit, let us also walk in the Spirit. (Galatians 5:25)The appeal to the believer is to produce fruit consistent with faith, as a consequence of being accepted in the Beloved (on His account), to be practically (in heart and deed) as they are positionally in Christ, to be as much conformed to the Lord Jesus in this life as we can be, and will be in the resurrection. (Philippians 3:7-21)However, this saving justifying faith, is a faith which effects obedience by the Spirit, in word and in deed, in heart and in life, whereby "the righteousness of the law might be fulfilled in us, who walk not after the flesh, but after the Spirit, (Romans 8:4) insofar as we do. And since faith and works go together like light and heat, sometimes they are used interchangeably as to what they effect. And which obedience includes penitent confession when convicted of not pleasing the Object of his faith for salvation, the risen Lord Jesus.Thus Purgatory is based upon a false premise, that of the need for perfection of character (if by grace) in order to be with God, versus penitent faith which appropriates justification, which purifies the heart (Acts 15:9) and is counted for righteousness (Romans 4:5) and renders one accepted in the Beloved (on His account) and positionally seated together with their Lord in Heaven, (Ephesians 1:6; 2:6) from where they await the Lord's return and His final subduing of our "vile body," that it may be fashioned like unto his glorious body," (Philippians 3:21)and which is the only transformative change after this life that the Scriptures speak of.