La proporción del tiempo X que un autómata industrial trabaja durante una semana de 40 horas es una variable aleatoria con la siguiente función de densidad de probabilidad: \(f(x) = \begin{cases} 2x & 0 \le x \le 1\\ 0 & \text{para las demás } x \end{cases}\) a) Encuentre E(X) y V(X). b) La ganancia semanal Y para este autómata está dada por Y = 200 X - 60, determine E(Y) y V(Y).
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The proportion of time $Y$ that an industrial robot is in operation during a 40 -hour week is a random variable with probability density function $$f(y)=\left\{\begin{array}{ll} 2 y, & 0 \leq y \leq 1 \\ 0, & \text { elsewhere } \end{array}\right.$$ a. Find $E(Y)$ and $V(Y)$ b. For the robot under study, the profit $X$ for a week is given by $X=200 Y-60$. Find $E(X)$ and $V(X)$ c. Find an interval in which the profit should lie for at least $75 \%$ of the weeks that the robot is in use.
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