Some authors modify our definition of a geometric random...

Some authors modify our definition of a geometric random variable $X$ so that it counts the number of trials including the trial at which the first success occurs. Thus, $X$ can assume the values $1,2,3, \ldots$. For this modified geometric random variable, find the pmf $p(\cdot)$, the expected value, and the variance in terms of the probability of success on each trial $p$ and of $q=1-p$.

Key Concepts

Variance

Variance measures the spread or dispersion of a random variable's distribution around its mean. For the geometric distribution, the variance is derived as Var(X) = q/p^2, where q is the failure probability (1-p). This formula explains how the uncertainty in the number of trials until the first success increases as the likelihood of success on each trial decreases.

Expected Value

The expected value of a random variable provides a measure of its central tendency or average outcome. For the modified geometric distribution, the expected value is calculated using the formula E(X) = 1/p, meaning that on average, one would expect the first success to occur after 1/p trials. This result arises from summing over all possible trials weighted by their probabilities.

Geometric Distribution

The geometric distribution is a discrete probability distribution that models the number of independent Bernoulli trials needed to obtain the first success. In its modified form, it counts the trial on which the first success occurs, meaning its support starts at 1 rather than 0. This distribution is memoryless, indicating that the probability of success in future trials is independent of past failures.

Probability Mass Function (pmf)

The probability mass function (pmf) for the modified geometric distribution assigns probabilities to each possible outcome. It is expressed as P(X = x) = q^(x-1) * p, where x is a positive integer (x = 1, 2, 3, ...), p is the probability of success on each trial, and q = 1 - p is the probability of failure. This function quantifies the probability that the first success occurs exactly on the x-th trial.

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