Question

The probability distribution of a discrete random variable X is

If $\text{a} = \text{P}(x<3)$ and $\text{b} = \text{P}(2 \le \text{X}<4)$, then

• A. $\text{a} = \text{b}$
• B. $a>b$
• C. $\text{a}<\text{b}$
• D. $\text{a} = \frac{1}{2} \text{b}$

Hint:

Always normalize probability first before calculating required values.

Answer 1

The Correct Option is C

Concept:
Sum of all probabilities = 1. Step 1: Find $k$. \[ 2k + k + 2k + 4k + k = 10k = 1 \Rightarrow k = \frac{1}{10} \] 
Step 2: Find $a = P(x<3)$. \[ a = P(0)+P(1)+P(2) = 2k + k + 2k = 5k = \frac{5}{10} = \frac{1}{2} \] 
Step 3: Find $b = P(2 \le X<4)$. \[ b = P(2)+P(3) = 2k + 4k = 6k = \frac{6}{10} = \frac{3}{5} \] 
Step 4: Compare. \[ \frac{1}{2}<\frac{3}{5} \Rightarrow a<b \] Conclusion. $a<b$