random variable

Question

Answer 1

$\begin{align*} &P(X = x) = K.\left ( 1- \beta \right )^{x-1} \quad \text{where x = } 1,2,3,4.... \infty \\ &\Rightarrow \text{ Above is a discrete probability distribution function} \\ &\Rightarrow \sum_{x=1}^{\infty} \left [ K.\left ( 1- \beta \right )^{x-1} \right ] = 1 \\ &\Rightarrow K. \sum_{x=1}^{\infty} \left [ \alpha^{x-1} \right ] = 1 \qquad \;\; {\color{red}{\alpha = 1-\beta }}\\ &\Rightarrow K. \sum_{x=1}^{\infty} \left [ \alpha^{1-1} + \alpha^{2-1} + \alpha^{3-1}...+ \infty \right ] = 1 \\ &\Rightarrow K.\left [ \frac{1}{1-\alpha} \right ] = 1 \\ &\Rightarrow K.\beta^{-1} = 1 \\ &\Rightarrow K = \beta \\ \end{align*}$