a partially discrete, partially continuous distribution, using a probability density function (which is normally used to represent absolutely continuous distributions). For example, the probability density function f(x) of a discrete distribution consisting of points x = {x1, ..., xn}, with corresponding probabilities p1, ..., pn, can be written as As another example, consider a distribution in which 6/10 of the time returns a standard normal distribution, and 4/10 of the time returns exactly the value 3.5 (i.e. a partly continuous, partly discrete mixture distribution). The density function of this distribution can be written as The delta function is also used to represent the resulting probability density function of a random variable that is transformed by continuously differentiable function. If Y = g(X) is a continuous differentiable function, then the density of Y can be written as The delta function is also used in a completely different way to represent the local time of a diffusion process (like Brownian motion). The local time of a stochastic process B(t) is given by and represents the amount of time that the process spends at the point x in the range of the process. More precisely, in one dimension this integral can be written where is the indicator function of the interval . Quantum mechanics The delta function is expedient in quantum mechanics. The wave function of a particle gives the probability amplitude of finding a particle within a given region of space. Wave functions are assumed to be elements of the Hilbert space L2 of square-integrable functions, and the total probability of finding a particle within a given interval is the integral of the magnitude of the wave function squared over the interval. A set {} of wave functions is orthonormal if they are normalized by where is the Kronecker delta. A set of orthonormal wave functions is complete in the space of square-integrable functions if any wave function can be expressed as a linear combination of the {} with complex coefficients: with . Complete orthonormal systems of wave functions appear naturally as the eigenfunctions of the Hamiltonian (of a bound system) in quantum mechanics that measures the energy levels, which are called the eigenvalues. The set of eigenvalues, in this case, is known as the spectrum of the Hamiltonian. In bra–ket notation, as above, this equality implies the resolution of the identity: Here the eigenvalues are assumed to be discrete, but the