Determine the mean and variance of a discrete random variable, given its distribution as follows.

The probability density function of a random variable X is f(x) = ke^-|x|, -∞ < x < ∞ Find the value of k and also find mean and variance for the random variable.

Let X be a random variable with cumulative distribution function
\(F(x)=\left\{\begin{array}{l} 0, \text { if } x<0 \\ \frac{x}{8}, \text { if } 0 \leq x<1 \\ \frac{1}{4}+\frac{x}{8}, \text { if } 1 \leq x<2 \\ \frac{3}{4}+\frac{x}{12}, \text { if } 2 \leq x<3 \\ 1, \text { for } 3 \leq x \end{array}\right.\)
(a) Compute: (i) P(1\(\le\)X\(\le\)2) and 
(ii) P(X=3)
(b) Is X a discrete random variable? Justify your answer.

The probability function of a random variable X is given by
\(p(x)=\left\{\begin{array}{l} \frac{1}{4}, \text { for } x=-2 \\ \frac{1}{4}, \text { for } x=0 \\ \frac{1}{2}, \text { for } x=10 \\ 0, \text { elsewhere } \end{array}\right.\)
Evaluate the following probabilities.
P(X<0)

The probability function of a random variable X is given by
\(p(x)=\left\{\begin{array}{l} \frac{1}{4}, \text { for } x=-2 \\ \frac{1}{4}, \text { for } x=0 \\ \frac{1}{2}, \text { for } x=10 \\ 0, \text { elsewhere } \end{array}\right.\)
Evaluate the following probabilities.
 P(0\(\le\)X\(\le\)10)