Question

The c.d.f. of a discrete random variable \( X \) is Then \( P(X = -3) / P(X \leq 0) \) is:

• A. \( \frac{1}{4} \)
• B. \( \frac{1}{6} \)
• C. \( \frac{1}{7} \)
• D. \( \frac{1}{8} \)

Hint:

To calculate the ratio of probabilities, first find the individual probabilities by summing up the relevant values from the c.d.f. Then, simply compute the ratio. Remember that the c.d.f. gives the cumulative probability up to a given value.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
For a discrete random variable, the probability at a point is \( P(X=x_i) = F(x_i) - F(x_{i-1}) \). \( P(X<0) \) is the sum of probabilities of values less than 0.
Step 2: Key Formula or Approach:
1. \( P(X = -3) = F(-3) = 0.1 \).
2. \( P(X<0) = P(X = -3) + P(X = -1) \). This is simply the c.d.f. value just before \( X=0 \).
Step 3: Detailed Explanation:
\( P(X = -3) = 0.1 \).
\( P(X<0) \) includes values \( -3 \) and \( -1 \).
From the table, \( F(-1) = P(X \le -1) = 0.3 \).
Since there are no values between -1 and 0, \( P(X<0) = P(X \le -1) = 0.3 \).
Calculation:
\( \frac{P(X = -3)}{P(X<0)} = \frac{0.1}{0.3} = \frac{1}{3} \).
Step 4: Final Answer:
The ratio is \( 1/3 \).