Continuous Probability Distributions
\(X\) is a random variable that follows a continuous uniform distribution with probability density function \[f(x) = \left\{\begin{matrix} \frac{1}{12} & (5 \leq x \leq 17) \\ 0 & (\text{elsewhere}). \end{matrix}\right. \] Then what is the mean of the distribution?
\(x\) is a random variable that follows a uniform distribution with the following probability density function: \[f(x) = \left\{\begin{matrix} \frac{1}{14} & (a \leq x \leq b) \\ 0 & (\text{elsewhere}). \end{matrix}\right. \] If the value of \(a+b\) is \(20,\) what is the value of \( a \times b?\)
\(X\) is a random variable that has a continuous uniform distribution with the probability density function \[f(x) = \left\{\begin{matrix} \frac{1}{26} & (5 \leq x \leq 31) \\ 0 & (\text{elsewhere}). \end{matrix}\right. \] Then what is the variance of the distribution?
\(x\) is a random variable that has a uniform distribution in the interval \([1,7].\) What is the probability of the random variable \(x\) taking on values greater than \(4?\)