Probability distributions While there are many probability distributions there are only a couple that are common used Gaussian f(x) (x-)2(2a2 a√2兀 -(x-)V-(x-) Multivariant f(x) (2) Chi-squared x (x) r(r12)22 03/1203 12540Lec10 Probability distributions The chi-squared distribution is the sum of the squares of r Gaussian random variables with expectation 0 and variance 1 With the probability density function known, the probability of events occurring can be determined For Gaussian distribution in 1-D; P(lxk<1o)=0.68 P(lx<2o)=0.955;P(x<3o)=0.9974 Conceptually, people thing of standard deviations in terms of probability of events occurring(ie. 68% of values should be within 1-sigma)03/12/03 12.540 Lec 10 19 Probability distributions • Gaussian f (x) = 1 s 2p e-(x-m) 2 s 2 ) f (x) = 1 (2p) n V e -1 2 (x-m) T V-1 (x-m) Chi - squared cr 2 (x) = xr/ 2-1 e-x / 2 G(r/ r/ 2 • While there are many probability distributions there are only a couple that are common used: /(2 Multivariant 2)2 03/12/03 12.540 Lec 10 20 Probability distributions • and variance 1. • With the probability density function known, the probability of events occurring can be determined. For Gaussian distribution in 1-D; P(|x|<1s) = 0.68; P(|x|<2s) = 0.955; P(|x|<3s) = 0.9974. • Conceptually, people thing of standard deviations in terms of probability of events occurring (ie. 68% of values should be within 1-sigma). The chi-squared distribution is the sum of the squares of r Gaussian random variables with expectation 0 10