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# (Solved): a) Suppose that $$X$$ is a continuous random variable with a probability density function: $f(x ... a) Suppose that $$X$$ is a continuous random variable with a probability density function: \[ f(x)=\left\{\begin{array}{ll} C\left(4 x-2 x^{2}\right) & 01) \). b) Let $$X$$ be a random variable. $$X$$ has the following probability distribution: What is the probability distribution of $$W=X^{2}$$ ? (c) The probability density function of $$X$$, the lifetime of a certain type of electronic device (measured in hours), is given by \[ f(x)=\left\{\begin{array}{ll} \frac{10}{x^{2}} & x \geq 10 \\ 0 & x \leq 10 \end{array}\right.$ i) Determine the cumulative distribution function of $$X$$. ii) Determine the mode (point of maximum density), and the median ( $$m$$ such that $$P(X \leq m)=F(m)=0.5)$$. iii) What is the probability that of 6 such types of devices at least 5 of them will function for at least 20 hours? iv) Let $$Y=\log (X)$$. Obtain the probability density function of $$Y$$.

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