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The area under the pdf curve is 1. Cumulative Distribution Function The cumulative distribution function, (CDF), is also known as the Probability Distribution Function, (PDF). to reduce confusion with the pdf of a random variable, we will use the acronym CDF to denote this function. The CDF of a random variable is the function defined by: The CDF and the pdf of a random variable are related: The CDF is the function corresponding to the probability that a given value x is less then the value of the random variable X.

Null Function The null functions of L2 are the set of all functions φ in L2 that satisfy the equation: for all a and b. Norm The L2 norm is defined as follows: [L2 Norm] If the norm of the function is 1, the function is normal. We can show that the derivative of the norm squared is: Scalar Product The scalar product in L2 space is defined as follows: [L2 Scalar Product] If the scalar product of two functions is zero, the functions are orthogonal. We can show that given coefficient matrices A and B, and variable x, the derivative of the scalar product can be given as: We can recognize this as the product rule of differentiation.

Banach Space A Banach Space is a complete normed function space. Hilbert Space A Hilbert Space is a Banach Space with respect to a norm induced by the scalar product. That is, if there is a scalar product in the space X, then we can say the norm is induced by the scalar product if we can write: That is, that the norm can be written as a function of the scalar product. In the L2 space, we can define the norm as: If the scalar product space is a Banach Space, if the norm space is also a Banach space.

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