The cumulative distribution function of a discrete random variable X is given. Then $\frac{P(X \le 0)}{P(X > 0)} = $ ______.
Show Hint
Never recalculate individual Probability Mass Function (PMF) values if you are given the Cumulative table! $P(X \le k)$ is just a straight lookup: it is simply the number printed exactly under $k$.
Step 1: Understanding the Question:
We are given a table displaying the Cumulative Distribution Function (CDF), denoted as $F(x)$, for a discrete random variable. We must find the ratio of two specific probability regions. Step 2: Detailed Explanation:
By the mathematical definition of a Cumulative Distribution Function:
$F(x) = P(X \le x)$
Looking at the provided table, we can directly extract the required probabilities:
1. Calculate the Numerator: $P(X \le 0)$
According to the definition, this is simply the value of the CDF at $x = 0$.
From the table, under $x = 0$, we find:
$F(0) = 0.5$
Therefore, $P(X \le 0) = 0.5$.
2. Calculate the Denominator: $P(X > 0)$
The sum of all probabilities in a valid distribution must exactly equal 1.
Using the complement rule of probability:
$P(X > 0) = 1 - P(X \le 0)$
$P(X > 0) = 1 - 0.5 = 0.5$
3. Calculate the Ratio:
$\text{Ratio} = \frac{P(X \le 0)}{P(X > 0)}$
$\text{Ratio} = \frac{0.5}{0.5} = 1$ Step 3: Final Answer:
The ratio is 1, matching option (b).
