Refer to Exercise 3.86 . The maximum likelihood estimator...

Refer to Exercise $3.86 .$ The maximum likelihood estimator for $p$ is $1 / Y$ (note that $Y$ is the geometric random variable, not a particular value of it). Derive $E(1 / Y) .[\text { Hint: If }|r|<1$, $\left.\sum_{i=1}^{\infty} r^{i} / i=-\ln (1-r) .\right]$.

Key Concepts

Geometric Distribution

The geometric distribution is a discrete probability distribution that models the number of independent Bernoulli trials needed to achieve the first success. Its probability mass function is typically expressed in terms of a parameter representing the success probability. This distribution is essential in understanding processes where the event of interest occurs for the first time after a sequence of failures, and it exhibits the memoryless property which simplifies various computations in probability and statistics.

Maximum Likelihood Estimation

Maximum Likelihood Estimation (MLE) is a method used to estimate the parameters of a statistical model by maximizing the likelihood function, which represents the probability of the observed sample given the parameters. In the context of distributions such as the geometric distribution, MLE provides a procedure to derive estimators that often have desirable properties like consistency and efficiency, making it a fundamental concept in statistical inference.

Expectation of a Function of a Random Variable

The expectation of a function of a random variable involves computing the average or mean value of the function when applied to the random variable. Instead of calculating the expectation of the random variable itself, this concept extends to functions of the variable, requiring the use of summation or integration (depending on whether the variable is discrete or continuous) over its distribution. This is especially useful when dealing with transformations like the reciprocal of a random variable.

Series Summation Identity

Series summation identities, such as the formula ?_{i=1}? r^i/i = -ln(1-r) for |r| < 1, are analytical tools that allow the evaluation of infinite series sums in closed forms. These identities are particularly useful in probability and statistics when calculating expectations or moments that involve nontrivial summations. They help simplify expressions and provide exact solutions where direct summation might be challenging.

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