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GO Classes CS/DA Test Series 2025 | Probability and Statistics | Topic Wise Test 3 | Question: 1

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Given the probability mass function $p(X=x)$ defined as:
$$
p(X=x)= \begin{cases}c\left(\frac{2}{3}\right)^{x} & \text { if } x=1,2,3, \ldots, \infty \\ 0 & \text { otherwise }\end{cases}
$$
find the constant $c$ so that $p(X=x)$ is a valid pmf for the random variable $X$.
  1. $c=\frac{1}{3}$
  2. $c=\frac{3}{2}$
  3. $c=\frac{1}{2}$
  4. $c=\frac{2}{3}$
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$$
\sum_{x=-\infty}^{\infty} p(x)=\sum_{x=1}^{\infty} c\left(\frac{2}{3}\right)^{x}=c \frac{2 / 3}{1-2 / 3}=2 c \quad \Rightarrow \quad c=\frac{1}{2}
$$
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