Discrete Mathematics
# 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 $x.$

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 $x.$

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 $x.$

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 $x.$