Question

\[ \text { The c.d.f. of a discrete random variable } \mathrm{X} \text { is } \]

\[ \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline \mathrm{X} & -3 & -1 & 0 & 1 & 3 & 5 & 7 & 9 \\ \hline \mathrm{~F}(\mathrm{X}=x) & 0.1 & 0.3 & 0.5 & 0.65 & 0.75 & 0.85 & 0.90 & 1 \\ \hline \end{array} \]

Then \frac{P[X=-3]}{P[X<0]}=$

• A. \(\frac{1}{4}\)
• B. \(\frac{1}{3}\)
• C. \(\frac{1}{6}\)
• D. \(\frac{1}{7}\)

Hint:

P(X = x) is the jump in the c.d.f. at x, and P(X below 0) is the c.d.f. just before 0.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
For a discrete variable, \(P(X=x_{i})=F(x_{i})-F(x_{i-1})\) and \(P(X \lt 0)\) is the sum of the probabilities at values less than 0.

Step 2: Find P(X = -3)
-3 is the smallest value, so \(P(X=-3)=F(-3)=0.1\).

Step 3: Find P(X below 0)
The values below 0 are -3 and -1, so \(P(X \lt 0)=F(-1)=0.3\).

Step 4: Ratio
\(\dfrac{P(X=-3)}{P(X \lt 0)}=\dfrac{0.1}{0.3}=\dfrac{1}{3}\).

Final Answer:
The ratio is 1/3. \[ \boxed{\text{(B) }\dfrac{1}{3}} \]