Question

If the p.m.f. of a random variable \( X \) is given by \[ P(X = x) = \frac{5}{25} \quad \text{if} \quad x = 0, 1, 2, 3, 4, 5 \] then which of the following is not true?

• A. \( P(X \leq 1) = P(X \geq 4) \)
• B. \( P(X \leq 2) \geq P(X \geq 4) \)
• C. \( P(X \leq 3) \leq P(X \geq 3) \)
• D. \( P(X \leq 2) = P(X \geq 3) \)

Hint:

When working with probability mass functions, make sure to check each probability condition carefully to ensure consistency.

Answer 1

The Correct Option is C

Step 1: Understanding the probabilities.
The probability mass function \( P(X = x) = \frac{5}{25} = \frac{1}{5} \) for \( x = 0, 1, 2, 3, 4, 5 \), so the total probability sum is 1. We calculate the probabilities for different ranges of \( X \).

Step 2: Checking the conditions.
We find that the statement \( P(X \leq 3) \leq P(X \geq 3) \) is false because both probabilities are equal, not less. Thus, option (C) is not true.

Step 3: Conclusion.
Thus, option (C) is the correct answer.