16. 322 Stochastic Estimation and Control. Fall 2004 Prof. VanderⅤelde P(a<X≤b)=P(X≤b)-P(X≤a) F(b-F(a) ∫f(x)ltx-「f(x)hx If(x)dx F(∞)=|f(a)d The density function is always a non-negative function, which integrates over the full ra Consider the probability that an observation lies in a narrow interval between x and x+ imnP(x<X≤x+dx)=lm∫faoh lim f(x)dx This can be used to measure f(x) We see clearly from this that the probability of a random variable having a continuous distribution function taking any prescribed value(dx>0)is zero In the case of a continuous random variable, the probability that it takes any exact value is 0 Also we shall use this relation in setting up problem solutions to indicate the probability of x taking a value in an infinitesimal interval near x f(x)=lim P(x<X<x+Ar) The appropriate range has to be chosen due to the fact that if you make intervals too small you'll have too few samples lying in each interval. The sampling error is too great if the number of samples you get in each interval is too small. If you make the interval too large then the approximation made in the limit is no longer pplicable. Discrete variable Page 3 of 716.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde P a( < X ≤ b) = PX ( ( ≤ b) − PX ≤ a) = Fb () − Fa ( ) b a f x dx () () − ∫ = f x dx ∫ −∞ −∞ b = f x dx ( ) ∫ a ∞ F( ) () ∞ = ∫ f u du = 1 −∞ The density function is always a non-negative function, which integrates over the full range to 1. Consider the probability that an observation lies in a narrow interval between x and x+dx. x dx + lim P x ( < X ≤ x + dx) = lim f u du ( ) dx→0 dx→0 ∫ x = lim f x dx ( ) dx→0 This can be used to measure f(x). We see clearly from this that the probability of a random variable having a continuous distribution function taking any prescribed value ( dx → 0 ) is zero. In the case of a continuous random variable, the probability that it takes any exact value is 0. Also we shall use this relation in setting up problem solutions to indicate the probability of x taking a value in an infinitesimal interval near x. f x) = lim P x < X ≤ x + ∆x) ( ( ∆ →x 0 ∆x The appropriate range has to be chosen due to the fact that if you make intervals too small you’ll have too few samples lying in each interval. The sampling error is too great if the number of samples you get in each interval is too small. If you make the interval too large then the approximation made in the limit is no longer applicable. Discrete variable Page 3 of 7