STAT 400, Fall '03 SOME PROBABILITY APPROXIMATIONS RELATED TO THE BINOMIAL ======================================================= A. Hypergeometric Approximations -------------------------------- DISTRIBUTION FUNCTION VALUES Binom(50, 0.4) Hyp(10000,4000,50) Hyp(400,160,50) Hyp(200,80,50) k 15 0.0955 0.0949 0.0811 0.0656 20 0.5610 0.5611 0.5642 0.5683 25 0.9427 0.9431 0.9542 0.9660 30 0.9986 0.9987 0.9993 0.9997 35 1.0 1.0 1.0 1.0 B. Poisson Approximations ------------------------- DISTRIBUTION FUNCTION VALUES Poisson(5) Binom(80,0.0625) Binom(100, 0.05) Binom(500,0.01) k 0 0.0067 0.0057 0.0059 0.0066 1 0.0404 0.0363 0.0371 0.0398 2 0.1247 0.1166 0.1183 0.1234 5 0.6160 0.6160 0.6160 0.6160 7 0.8666 0.8735 0.8720 0.8677 8 0.9319 0.9382 0.9369 0.9329 9 0.9682 0.9727 0.9718 0.9689 10 0.9863 0.9891 0.9885 0.9868 SOME SIMULATED RANDOM EXPERIMENTS WITH EXPECTED VALUES (1) Binomial with n=4, p=.3. True expectation of such a random variable is n*p = 1.2. Batch 1: simulate 1000 Binom(4,.3) r.v.'s, giving total # successes = 1152, or `empirical average' 1.152 Batch 2: again simulate 1000 such r.v.'s. The true probability mass function is: 0.2401 0.4116 0.2646 0.0756 0.0081 probabilities for respective values 0,1,2,3,4. In other words the EXPECTED numbers out of 1000 batches for which the r.v. values were respectively 0,..,4 were: Value 0 1 2 3 4 Exp.# 240.1 411.6 264.6 75.6 8.1 Actual# 245 384 285 75 11 Expected number total successes = 240.1*0+ 411.6*1+ 264.6*2+ 75.6*3+ 8.1*4 = 1200 Actual number of total successes = 1223 Batch 3: again 1000 r.v.'s, Binom(1,.6). Expected total 600, observed 594. Batch 4: another 1000 like Batch 3: observed total 640. MANY MORE SIMULATION EXPERIMENTS WITH THIS FLAVOR WILL BE REPORTED IN FUTURE HANDOUTS !!!