16. 322 Stochastic Estimation and Control, Fall 2004 Prof vander velde n Useful relations involving sums of binomial coefficients n +.+=2", n a positive integer 0)(1 +…±=0, n a positive integer n 4. The mean, mean square and variance of the binomial distribution s)=∑Ps ∑|(msyq (ps+q The moments can be derived from this using n(ps+qp ds n(n-D)(ps+q dG dgl dG n(n X2-F2 =n npq Page 7 of 1016.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde ⎛ ⎞ n ⎜ ⎟ = 1 ⎝ ⎠ 0 ⎛ ⎞ n ⎜ ⎟ = 1 ⎝ ⎠ n Useful relations involving sums of binomial coefficients: ⎛ n ⎞ ⎛ ⎞ n ⎛ n +1⎞ + = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟, n any number k ⎝ k −1⎠ ⎝ ⎠ ⎝ k ⎠ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ n n n n ⎜ ⎟ ⎜ ⎟ + + ⎜ ⎟ + ... = 2 , n a positive integer ⎝ ⎠ 0 1 ⎝ ⎠ ⎝ ⎠ n ⎛ ⎞ n n n ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ + ± ⎜ ⎟ − ... = 0, n a positive integer ⎝ ⎠ 0 1 ⎝ ⎠ ⎝ ⎠ n 4. The mean, mean square, and variance of the binomial distribution Gs k k () = ∑ p s k n ⎛ ⎞ n k n−k k = ∑⎜ ⎟p q s k =0 k⎝ ⎠ n n ⎛ ⎞ k n−k k = ∑ ps q ⎜ ⎟() k =0 ⎝ ⎠ n = ( ps + q) The moments can be derived from this using dG = n ps ( + q) n−1 p ds 2 dG ( 2 = nn −1)( ps + q) n−2 p ds2 dG X = = np ds s=1 2 X 2 dG dG = + ds2 ds s=1 s=1 = nn ( −1) p 2 + np 2 2 σ = X − X 2 2 2 2 2 = n p − np 2 + np − n p = npq Page 7 of 10