For each of the following, find the constant c so that p(x)...

For each of the following, find the constant $c$ so that $p(x)$ satisfies the condition of being a pmf of one random variable $X$.
(a) $p(x)=c\left(\frac{2}{3}\right)^{x}, x=1,2,3, \ldots$, zero elsewhere.
(b) $p(x)=c x, x=1,2,3,4,5,6$, zero elsewhere.

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Key Concepts

Geometric Series

A geometric series is used to sum an infinite sequence where each term is a constant multiple of the previous term. In problems involving pmfs that are defined on an infinite set, the sum of probabilities often forms a geometric series, which can be summed using the formula S = a/(1 - r) for |r| < 1.

Finite Series

A finite series occurs when the random variable takes on a finite set of values. The sum of probabilities is obtained by directly summing a finite number of terms. In such cases, the normalization constant is determined by setting the finite sum equal to 1 and solving the resulting equation.

Probability Mass Function

A probability mass function (pmf) assigns a probability to each outcome in the sample space of a discrete random variable. It must satisfy two conditions: each probability is non-negative and the sum of all probabilities equals 1.

Normalization Condition

The normalization condition ensures that the total probability over all outcomes of the random variable is 1. When a pmf includes an unknown constant, this condition is used to solve for the constant by setting the sum of the probabilities equal to 1.

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