Question

Let \( X \) be a random variable having distribution function

\[ F(x) = \begin{cases} 0, & x<1 \\ a, & 1 \leq x<2 \\ \frac{c}{2}, & 2 \leq x<3 \\ 1, & x \geq 3 \end{cases} \]

where \( a \) and \( c \) are appropriate constants. Let \( A_n = \left[ 1 + \frac{1}{n}, 3 - \frac{1}{n} \right] \), \( n \geq 1 \), and \( A = \bigcup_{i=1}^{\infty} A_i \). If \( P(X \leq 1) = \frac{1}{2} \) and \( E(X) = \frac{5}{3} \), then \( P(X \in A) \) equals _________ (round off to 2 decimal places).

Hint:

To solve for probabilities in such distribution problems, use the properties of the cumulative distribution function and apply the given information systematically.

Answer 1

Correct Answer: 0.32

We are given the distribution function and the probabilities \( P(X \leq 1) = \frac{1}{2} \) and \( E(X) = \frac{5}{3} \). From the distribution function, we can solve for the values of \( a \) and \( c \). Using the properties of the cumulative distribution function and the expected value formula, we can determine that: \[ P(X \in A) = 0.32. \] Thus, the value is \( 0.32 \).