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In summary, it is possible to find the CDF of a random variable directly from its characteristic function without first finding the PDF through inverse Fourier transform. This can be done by integrating the expression for F(x) from a to b, where b is the desired CDF and a tends to -∞. This is possible because the same expression holds even when there is no PDF available.

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mathman

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mathman said:

Sorry, this isn't clear to me. Could you elaborate more? Where is ##x## in the integration to have ##F(x)##?

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mathman

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[tex]F(x)=\int_{-\infty}^{\infty}\frac{e^{-itx}}{-it}\phi(it)\,dt[/tex]

which is the IFT of ##\frac{\phi(it)}{-it}##?

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mathman

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∞9t

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.

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