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Is there a way to find the CDF of a random variable from its characteristic function directly, without first finding the PDF through inverse Fourier transform, and then integrate the PDF to get the CDFÉ
mathman said:Let F(x) be the desired cdf. You can get [tex]F(b)-F(a)=\frac{1}{2\pi}\int_{-\infty}^{\infty} \frac{e^{-itb}-e^{-ita}}{-it}\phi(t)dt[/tex]$.
The CDF (cumulative distribution function) from the Characteristic Function is a mathematical function that describes the probability of a random variable being less than or equal to a specific value. It is obtained by taking the inverse Fourier transform of the Characteristic Function, which is the Fourier transform of the probability density function.
The CDF is the integral of the PDF (probability density function) over a given range of values. It represents the probability of a random variable falling within that range. The PDF, on the other hand, is the derivative of the CDF and describes the probability distribution of a continuous random variable.
The CDF is a fundamental concept in statistics as it allows us to calculate the probability of a random variable taking on a specific value or falling within a certain range of values. It is also useful in calculating other important statistical measures such as the mean, median, and variance.
Yes, the CDF can be used for both continuous and discrete random variables. For a continuous random variable, the CDF is a smooth curve, while for a discrete random variable, it is a step function. In both cases, the CDF provides information about the probability distribution of the random variable.
The characteristic function is the Fourier transform of the probability density function, while the CDF is the inverse Fourier transform of the characteristic function. This means that the CDF and the characteristic function are closely related, and both contain information about the probability distribution of a random variable.