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: Recall the cumulative distribution function.
For a random variable $X$, $F(x)=P(X\le x)$. We want a clean expression for $F(0)=P(X\le 0)$.
Step 2: Split the whole sample space at $0$.
Every outcome satisfies exactly one of $X\le 0$ or $X>0$; these two events are mutually exclusive and together cover all possibilities.
Step 3: Use the total probability axiom.
Therefore $P(X\le 0)+P(X>0)=1$.
Step 4: Isolate $F(0)$.
\[ F(0)=P(X\le 0)=1-P(X>0). \]
Step 5: Why the complement form is preferred.
This identity holds for any distribution, discrete or continuous, regardless of the exact entries in the probability table, so it is the safe general answer.
Step 6: Conclude.
Matching the options, $F(0)=1-P(X>0)$. \[ \boxed{F(0)=1-P(X>0)} \]