16.322 Stochastic Estimation and Control, Fall 2004 Prof vander Velde he limiting distribution of S is the normal distribution Note that this is true for any distributions of the Xi These are sufficient conditions under which the theorem can be proved. It is not lear that they ar Notice that each of the noises mentioned earlier depend on the accumulated effect of a great many small causes e.g., voltage across plate: electrons traveling from cathode to plate It is convenient to work with the characteristic function since we are dealing with the sum of independent variables Normal probability density function f(x) /2 Normal probability distribution function 1。smt 2丌 √2z Where X This integral with the integrand normalized is tabulated. It is called the normal probability function and symbolized with p 9/30/2004955AM Page 4 of 1016.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde 9/30/2004 9:55 AM Page 4 of 10 the limiting distribution of S is the normal distribution. Note that this is true for any distributions of the Xi. These are sufficient conditions under which the theorem can be proved. It is not clear that they are necessary. Notice that each of the noises mentioned earlier depend on the accumulated effect of a great many small causes e.g., voltage across plate: electrons traveling from cathode to plate. It is convenient to work with the characteristic function since we are dealing with the sum of independent variables. Normal probability density function: 2 2 ( ) 2 1 ( ) 2 x m fx e σ π − − = Normal probability distribution function: 2 2 2 ( ) 2 2 1 ( ) 2 1 2 Where: 1 x u m x m v F x e du e dv m X u m v dv du σ σ πσ πσ σ σ − − −∞ − − −∞ = = = − = = ∫ ∫ This integral with the integrand normalized is tabulated. It is called the normal probability function and symbolized with Φ