The Fokker-Planck equation is a differential equation for the distribution function of Brownian motion, first derived by Fokker and Planck. It allows the calculation of expectation values for processes described by nonlinear Langevin equations, which is more complex than for linear ones. The derivation starts with the Kramers-Moyal expansion, which involves Kramers-Moyal coefficients. For Langevin equations with delta-correlated Gaussian forces, these coefficients vanish for $ n \geq 3 $, leaving only the drift and diffusion coefficients in the distribution function equation. This results in the Fokker-Planck equation, also known as the forward Kolmogorov equation. The problem of calculating averages is thus reduced to solving this equation. The derivation begins with the transition probability, which connects the probability density at two different times. By expanding the transition probability in terms of moments, the characteristic function is used to derive the transition probability. The expansion leads to an expression for the transition probability in terms of the moments. The first method uses the characteristic function to derive the transition probability, while the second method uses a Taylor series expansion of the delta function. The result is an expression for the transition probability in terms of the moments, which is then used to derive the Fokker-Planck equation. The derivation is presented for both one-variable and N-variable cases.The Fokker-Planck equation is a differential equation for the distribution function of Brownian motion, first derived by Fokker and Planck. It allows the calculation of expectation values for processes described by nonlinear Langevin equations, which is more complex than for linear ones. The derivation starts with the Kramers-Moyal expansion, which involves Kramers-Moyal coefficients. For Langevin equations with delta-correlated Gaussian forces, these coefficients vanish for $ n \geq 3 $, leaving only the drift and diffusion coefficients in the distribution function equation. This results in the Fokker-Planck equation, also known as the forward Kolmogorov equation. The problem of calculating averages is thus reduced to solving this equation. The derivation begins with the transition probability, which connects the probability density at two different times. By expanding the transition probability in terms of moments, the characteristic function is used to derive the transition probability. The expansion leads to an expression for the transition probability in terms of the moments. The first method uses the characteristic function to derive the transition probability, while the second method uses a Taylor series expansion of the delta function. The result is an expression for the transition probability in terms of the moments, which is then used to derive the Fokker-Planck equation. The derivation is presented for both one-variable and N-variable cases.