Question

Suppose that a random variableXhas a discrete distribution

with the following p.f.:

\(f\left( x \right) = \left\{ \begin{array}{l}\frac{c}{{{2^x}}}\;\;for\;x = 0,1,2,...\\0\;\;\;\;otherwise\end{array} \right.\)

Find the value of the constantc.

Short answer

The value of constant c is \(\frac{1}{2}\).

Step-by-step solution

Given information

The random variable X follows the discrete distribution.

The probability function is given as,

\(f\left( x \right) = \left\{ \begin{array}{l}\frac{c}{{{2^x}}}\;\;for\;x = 0,1,2,...\\0\;\;\;\;otherwise\end{array} \right.\)

Calculate the value of constant c

It is known that the sum of the probabilities for all events in the sample space of an experiment is equal to 1.

From the provided probability function, the value for constant c is computed as,

\(\begin{aligned}{c}\sum\limits_i {f\left( x \right)} &= 1\\\sum\limits_{x = 0}^\infty {\frac{c}{{{2^x}}}} &= 1\\\left( {c + \frac{c}{2} + \frac{c}{4} + \frac{c}{8} + ...} \right)& = 1\\c\left( {1 + \frac{1}{2} + \frac{1}{{{2^2}}} + \frac{1}{{{2^3}}} + ...} \right)& = 1\end{aligned}\)

Since it’s an infinite series (geometric progression) with the first term as\({\bf{a = 1}}\)and the common ratio as\({\bf{r = }}\frac{{\bf{1}}}{{\bf{2}}}\), the sum of infinite series is obtained as,\(\frac{{\bf{a}}}{{{\bf{1 - r}}}}\).

Therefore,

\(\begin{aligned}{c}\frac{{c\left( 1 \right)}}{{1 - \frac{1}{2}}} &= 1\\2c& = 1\\c &= \frac{1}{2}\end{aligned}\)

Thus, the value of constant c is \(\frac{1}{2}\).