Let X have the pmf p(x)=((1)/(3))((2)/(3))^x, x=0,1,2,3, …, zero elsewhere. Find the conditional

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Let X have the pmf p(x)=((1)/(3))((2)/(3))^x, x=0,1,2,3, …,...

Key Concepts

Memoryless Property

The memoryless property states that for a geometric distribution, the probability of success in the future is independent of the past. That is, the process 'restarts' at every trial. This property is key in simplifying the calculation of conditional probabilities when the event condition involves a shift in the support, such as X being greater than or equal to a specific value.

Summation of a Geometric Series

To compute conditional probabilities accurately, especially when dealing with events defined over an infinite set of outcomes, it is important to be able to sum an infinite geometric series. This concept allows one to calculate the normalization constant (the probability of the conditioning event) for the conditional probability distribution.

Geometric Distribution

The geometric distribution models the number of trials needed to achieve the first success in a sequence of independent Bernoulli trials with constant success probability. In this example, the given pmf is a variant of the geometric distribution with support starting at 0. Understanding the properties of this distribution is crucial for solving problems involving waiting times and failure counts.

Conditional Probability

This concept involves finding the probability of an event occurring given that another event has already occurred. In problems like this, the conditional probability is calculated by dividing the probability of the joint event by the probability of the condition. In the context of probability mass functions, it involves adjusting the pmf so that it sums to one over the subset of interest.

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