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How many heartbeats do you have each minute? How many points will your favorite team score in their game tonight? These any many other real-world values can be modeled by discrete random variables.

If the cumulative distribution function of a discrete random variable \( X \) which takes on integer values is given by \[ F_X(x) = \begin{cases} 0 \quad (x\lt 0) \\ c \left\lfloor x \right\rfloor \quad ( 0 \le x \lt 12 )\\ 1 \quad (x \ge 12), \end{cases} \] which of the following is an impossible value for the probabilty \( P(X = 7)? \)

**Note:** \( \left\lfloor x \right\rfloor \) refers to the greatest integer equal to or smaller than \(x. \)

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If the cumulative distribution function of a discrete random variable \( X \) which takes on integer values is \[ F_X(x) = \begin{cases} 0 \quad (x\lt 0) \\ \frac{1}{10 ^{2}} \left( - \left\lfloor x \right\rfloor ^2 + 20 \left\lfloor x \right\rfloor \right) \quad ( 0 \le x \lt 10 )\\ 1 \quad (x \ge 10), \end{cases} \] what is the probabilty \( P(X = 4)? \)

**Note:** \( \left\lfloor x \right\rfloor \) refers to the largest integer not greater than \(x. \)

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If the cumulative distribution function of a discrete random variable \( X \) which takes on integer values is \[ F_X(x) = \begin{cases} 0 \quad (x\lt 0) \\ \frac{\left\lfloor x \right\rfloor ^2}{100} \quad ( 0 \le x \lt 10 )\\ 1 \quad (x \ge 10), \end{cases} \] what is the probabilty \( P( X \ge 4)? \)

**Note:** \( \left\lfloor x \right\rfloor \) refers to the largest integer not greater than \(x. \)

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If the cumulative distribution function of a discrete random variable \( X \) which takes on integer values is \[ F_X(x) = \begin{cases} 0 &\quad (x\lt 0) \\ \frac{\left\lfloor x \right\rfloor ^2}{144} &\quad ( 0 \le x \lt 12 )\\ 1 &\quad (x \ge 12) \end{cases} \] what is \( E[X]? \)

**Note:** \( \left\lfloor x \right\rfloor \) refers to the largest integer not greater than \(x. \)

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