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Note that, since x + y is a vector and α is a multi-index, the expression on the left is short for (x 1 + y 1) α 1 ⋯(x n + y n) α n. Leibniz formula For smooth functions f {\textstyle f} and g {\textstyle g} , ∂ α ( f g ) = ∑ ν ≤ α ( α ν ) ∂ ν f ∂ α − ν g . {\displaystyle \partial ^{\alpha }(fg)=\sum _{\nu \leq \alpha ...
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.According to the theorem, the power (+) expands into a polynomial with terms of the form , where the exponents and are nonnegative integers satisfying + = and the coefficient of each term is a specific positive integer ...
The name of a binomial process is derived from the fact that for all measurable sets the random variable follows a binomial distribution with parameters () and : ξ ( A ) ∼ Bin ( n , P ( A ) ) . {\displaystyle \xi (A)\sim \operatorname {Bin} (n,P(A)).}
The formula in the definition of characteristic function allows us to compute φ when we know the distribution function F (or density f). If, on the other hand, we know the characteristic function φ and want to find the corresponding distribution function, then one of the following inversion theorems can be used. Theorem.
The binomial approximation for the square root, + + /, can be applied for the following expression, + where and are real but .. The mathematical form for the binomial approximation can be recovered by factoring out the large term and recalling that a square root is the same as a power of one half.
Relationship to the binomial theorem [ edit ] The Leibniz rule bears a strong resemblance to the binomial theorem , and in fact the binomial theorem can be proven directly from the Leibniz rule by taking f ( x ) = e a x {\displaystyle f(x)=e^{ax}} and g ( x ) = e b x , {\displaystyle g(x)=e^{bx},} which gives
All covariances are negative because for fixed n, an increase in one component of a multinomial vector requires a decrease in another component. When these expressions are combined into a matrix with i, j element cov ( X i , X j ) , {\displaystyle \operatorname {cov} (X_{i},X_{j}),} the result is a k × k positive-semidefinite covariance ...
The Gaussian binomial coefficient, written as () or [], is a polynomial in q with integer coefficients, whose value when q is set to a prime power counts the number of subspaces of dimension k in a vector space of dimension n over , a finite field with q elements; i.e. it is the number of points in the finite Grassmannian (,).