Question: What conditions are necessary on a function \(f\) for it to be the probability density function of a continuous random variable? What are the probabilistic interpretations of these conditions? Why is \(f\) called a density function?

If \(X\) is a continuous random variable describe the relationship between the cumulative distribution function and the probability density function of \(f\). Suppose \(X\) is a non-negative continuous random variable with probability density function \(f(x)=C e^{-x}-e^{-2 x}\) for \(x \geq 0\), where \(C\) is a constant which you should find. Find the cumulative distribution function of \(X\). Find also the mean, median and mode of this distribution.

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