Question

The value of the constant \( c \), so that \( P(x) = c\left(\dfrac{2}{3}\right)^x,\, x = 1,2,3,\ldots \) is the probability distribution function of a discrete random variable \( X \), is:

• A. \(\dfrac{1}{3}\)
• B. \(\dfrac{1}{2}\)
• C. \(1\)
• D. \(2\)

Hint:

Use the sum of infinite geometric series: \( \sum_{n=1}^\infty ar^n = \dfrac{ar}{1 - r} \), valid for \( |r|<1 \).

Answer 1

The Correct Option is B

Given the probability mass function: \[ P(x) = c\left(\dfrac{2}{3}\right)^x, \quad x = 1,2,3,\ldots \] To find \( c \), use the condition that the total probability must sum to 1: \[ \sum_{x=1}^{\infty} P(x) = 1 \Rightarrow c \sum_{x=1}^{\infty} \left(\dfrac{2}{3}\right)^x = 1 \] This is a geometric series: \[ \sum_{x=1}^{\infty} \left(\dfrac{2}{3}\right)^x = \dfrac{\dfrac{2}{3}}{1 - \dfrac{2}{3}} = \dfrac{2/3}{1/3} = 2 \] Thus: \[ c \cdot 2 = 1 \Rightarrow c = \dfrac{1}{2} \] % Final Answer \[ \boxed{c = \dfrac{1}{2}} \]