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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 \( Y \).

(a) i) ?x=02C(4x?2x2)dx=1? C((2x2?(23)x3)02)=1 ? C=38 ii) P(X>1)=?12(38)×(4x?2x2)dx=12 (b) P(W=0)=P(X=0)=25 P(W=1)=P(X=1)+P(X=?1)=35 P(W=k) = 0 , ?k?0

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