Question

If the probability distribution function of a random variable $X$ is given as

Then $F(0)$ is equal to

• A. $P(X > 0)$
• B. $1 - P(X > 0)$
• C. $1 - P(X < 0)$
• D. $P(X < 0)$

Hint:

For any cumulative distribution problem, remember the complementary event shortcut: $P(X \le x) = 1 - P(X > x)$. This identity allows you to bypass summing individual columns directly by inspecting total probability balances.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The problem presents a discrete random variable $X$ with its associated probability distribution. We are required to find an alternate expression for the cumulative distribution function evaluated at zero, denoted as $F(0)$.

Step 2: Key Formula or Approach:
The cumulative distribution function $F(x)$ represents the sum of probabilities for all outcomes less than or equal to $x$: $$F(0) = P(X \le 0)$$ By the total probability axiom, the sum of all possible probabilities in a sample space is strictly equal to 1: $$P(X \le 0) + P(X > 0) = 1 \implies P(X \le 0) = 1 - P(X > 0)$$

Step 3: Detailed Explanation:
The complete set of outcomes is partitioned into two mutually exclusive, exhaustive events: outcomes where $X \le 0$ and outcomes where $X > 0$.
Therefore, we can establish the complementary probability relationship: $$F(0) = P(X \le 0) = 1 - P(X > 0)$$ Let's check the options: option (B) explicitly matches the form $1 - P(X > 0)$. This relationship holds true regardless of the specific numerical values filled inside the placeholder table.

Step 4: Final Answer:
The value of $F(0)$ is equivalent to $1 - P(X > 0)$, which corresponds to option (B).