To solve the problem, we need to find the expected value and variance of the random variable $S = \ln(R)$, given the probability mass function $p_R(r)$ of the random variable $R$.

Step 1: Understanding the Expected Value of $S = \ln(R)$

The expected value of $S$ is the weighted average of $\ln(R)$ values, where the weights are the probabilities given by the probability mass function $p_R(r)$. Mathematically, the expected value of $S$ is:

$\mathbb{E}(S) = \mathbb{E}(\ln(R)) = \sum_{r} \ln(r) \cdot p_R(r)$

where the sum is taken over all possible values of $r$ in the support of $R$.

Step 2: Finding the Variance of $S$

The variance of $S$ is calculated using the formula:

$\text{Var}(S) = \mathbb{E}(S^2) - (\mathbb{E}(S))^2$

First, compute $\mathbb{E}(S^2)$, which is:

$\mathbb{E}(S^2) = \mathbb{E}((\ln(R))^2) = \sum_{r} (\ln(r))^2 \cdot p_R(r)$

Then, subtract $(\mathbb{E}(S))^2$ from $\mathbb{E}(S^2)$ to obtain the variance.

Summary

1. Compute $\mathbb{E}(S) = \sum_{r} \ln(r) \cdot p_R(r)$.
2. Compute $\mathbb{E}(S^2) = \sum_{r} (\ln(r))^2 \cdot p_R(r)$.
3. Calculate the variance as $\text{Var}(S) = \mathbb{E}(S^2) - (\mathbb{E}(S))^2$.

Would you like to provide a specific probability mass function for $p_R(r)$, so we can continue with the actual computation?

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Related Questions:

1. How do you compute the expected value and variance of any transformed random variable?
2. What are some common transformations of random variables in probability theory?
3. How does the logarithmic transformation affect the distribution of a random variable?
4. What role does the probability mass function (PMF) play in calculating expected values?
5. How can variance be used to understand the spread of transformed random variables?

Tip: The expected value of a transformed random variable $g(X)$ can always be found using $\mathbb{E}(g(X)) = \sum_x g(x) \cdot p_X(x)$.