Suppose you choose at random a real number \(X\) from the interval \([2, 10]\). Find the density function \(f(x)\) and the probability of an event \(E\) for this experiment, where \(E\) is a subinterval \([a, b]\) of \([2, 10]\).
\[ f(x) = \frac{1}{b - a} \]
\(a = 2\) and \(b = 10\), so:
\[ f(x) = \frac{1}{10 - 2} \]
Density function \(f(x)\):
\[ f(x) = \begin{cases} \frac{1}{8}, & \text{if } 2 \leq x \leq 10 \\ 0, & \text{otherwise} \end{cases} \]
The probability of an event \(E\) for \([a, b]\) is given by:
\[ P(a \leq X \leq b) = \frac{1}{b - a} \]
In this case, \(a = 2\) and \(b = 10\), so the probability of the event \(E\) for the subinterval \([2, 10]\) is:
\[ P(2 \leq X \leq 10) = \frac{1}{10 - 2} \]
Therefore, the probability of the event \(E\) is:
\[ P(2 \leq X \leq 10) = \frac{1}{8} \]