(a) Show that p(n)=((1)/(2))^n+1 for n=0,1,2, …is a valid PMF for a discrete r.v. (b) Find the C

step 1 So we're given the probability is equal to 1 over n, and x could be any number from 1 to n. And

(a) Show that p(n)=((1)/(2))^n+1 for n=0,1,2, …is a valid...

Key Concepts

Discrete Random Variable

A discrete random variable is one that can take on a countable number of distinct values. Understanding discrete random variables is crucial because their probability distributions are characterized by individual probabilities assigned to these outcomes, often described by a probability mass function.

Probability Mass Function (PMF)

A probability mass function (PMF) assigns a probability to each outcome of a discrete random variable, while satisfying the conditions that each probability is non-negative and that the sum of the probabilities over all possible outcomes equals 1. This function is fundamental in describing the behavior of discrete random variables.

Validation of a PMF using a Geometric Series

To validate a PMF, one must confirm that it satisfies the two main criteria: non-negativity for all outcomes and that the total sum of the probabilities is 1. For functions expressed in the form of powers, like (1/2)^(n+1), the sum is evaluated using the formula for an infinite geometric series. This process confirms that the probabilities add up to 1, ensuring a valid PMF.

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) for a discrete random variable gives the probability that the variable is less than or equal to a particular value. It is obtained by summing the probabilities provided by the PMF up to that value. The CDF plays a key role in determining probabilities over ranges and in understanding the distribution's behavior in a cumulative context.

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