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Discrete Mathematics

# Continuous Random Variables - Probability Density Function (PDF)

If the probability density function of a continuous random variable $$X\in[-1,7]$$ is given by $f(x)=a(x+1)(x-7),$ what is $$P(0\le X\le4)?$$

The probability density function of a continuous random variable $$X$$ that ranges from 1 to $$e^{2}$$ is given by $f(x)=\frac{1}{2x}.$ Find the value of $$k$$ such that $$P(X>k)=\frac{1}{10}.$$

If the probability density function of a continuous random variable $$X$$ is given by $f(x)=\begin{cases} ax^2\ &(0\le x\le1) \\-\frac{a}{9}(x-1)+a\ &(1<x\le10),\end{cases}$ what is the value of $$a?$$

If the probability density function of a continuous random variable $$X\in[0,\frac{\pi}{16}]$$ is given by $f(x)=a\sin8x,$ what is the variance of $$X?$$

If the probability density function of a continuous random variable $$X$$ is given by $f(x)=11x^{10}\ (0\le x\le1),$ what is $$E[X]?$$

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