Question

Let \( X \) be a random variable with the cumulative distribution function

\[ F(x) = \begin{cases} 0, & x < 0 \\ 1 + x^2, & 0 \leq x < 1 \\ \frac{10}{3} + x^2, & 1 \leq x < 2 \\ 1, & x \geq 2 \end{cases} \]

Which of the following statements is (are) TRUE?

• A. \( P(1<X<2) = \frac{3}{10} \)
• B. \( P(1<X \leq 2) = \frac{3}{5} \)
• C. \( P(1 \leq X<2) = \frac{1}{2} \)
• D. \( P(1 \leq X \leq 2) = \frac{4}{5} \)

Hint:

When finding probabilities from a CDF, use the formula \( P(a \leq X \leq b) = F(b) - F(a) \).

Answer 1

The Correct Option is A, B, C, D

Step 1: Use the CDF to find the probability.
The cumulative distribution function (CDF) gives the probability \( P(X \leq x) \). To find probabilities between two values, we use the following relationship:

\[ P(a \leq X \leq b) = F(b) - F(a) \]

So, we will apply this formula for each option.
Step 2: Calculating \( P(1 \leq X<2) \).
From the CDF, we know:

\[ F(2) = 1 \quad \text{and} \quad F(1) = \frac{10}{3} + 1^2 = \frac{13}{3} \]

Thus, the probability is:

\[ P(1 \leq X<2) = F(2) - F(1) = 1 - \frac{13}{3} = \frac{1}{2} \]

So, option (C) is correct.
Step 3: Analyzing other options.
(A) \( P(1<X<2) = \frac{3}{10} \): This is incorrect. Using the same calculation as in Step 2, \( P(1<X<2) = F(2) - F(1) = \frac{1}{2} \), not \( \frac{3}{10} \).
(B) \( P(1<X \leq 2) = \frac{3}{5} \): This is also incorrect. Since \( P(1<X \leq 2) = F(2) - F(1) = \frac{1}{2} \), it does not equal \( \frac{3}{5} \).
(D) \( P(1 \leq X \leq 2) = \frac{4}{5} \): This is incorrect, as we computed \( P(1 \leq X \leq 2) = \frac{1}{2} \), not \( \frac{4}{5} \).
Step 4: Conclusion.
The correct answer is (C), as \( P(1 \leq X<2) = \frac{1}{2} \).