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In our exercise, the main character is the random variable we denote as \( X \). But what exactly is a random variable? Simply put, it's a variable that can take different outcomes based on a random phenomenon. Think of it as a way to map real-world occurrences into a mathematical framework. In this scenario, \( X \) represents the number of people who respond to the pollster when 20 cell phone numbers are dialed.
A random variable can be discrete or continuous. For this exercise, \( X \) is discrete because it has a specific set of possible values: 0, 1, 2, ..., up to 20, corresponding to the number of positive responses out of 20 attempts.
Understanding \( X \) allows us to translate a real-world process—people answering a poll—into a format that can be analyzed and used in probability calculations. This helps us predict outcomes and make informed decisions.

Probability plays a key role in understanding how likely certain outcomes are when dealing with random variables like \( X \). With our binomial distribution, we aim to figure out the probability of different numbers of responses, specifically using the binomial probability formula. This formula helps us compute the probability of achieving exactly \( k \) successes (in our case, responses) in \( n \) trials (calls) with a probability of \( p \) for each success.

Here's the formula again:
\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]
Where:

• \( \binom{n}{k} \) is the binomial coefficient, calculated as \( \frac{n!}{k!(n-k)!} \), which counts the number of ways to choose \( k \) successes from \( n \) trials.
• \( p^k \) gives the probability of \( k \) successes, while \( (1-p)^{n-k} \) accounts for the probability of \( n-k \) failures.

For instance, to figure out the likelihood of receiving responses from exactly two people, we insert \( n = 20 \), \( p = 0.1 \), and \( k = 2 \) into the formula, performing the calculations systematically. This helps us understand how probable our observations really are.

To fully grasp the behavior of a random variable, it's useful to understand both its expected value and its variance. These concepts give us insights into the average outcome we can anticipate and the variability of that outcome.

The expected value \( E(X) \), also known as the mean, indicates what we'd expect \( X \) to be on average. For a binomial distribution, it's calculated using the formula \( E(X) = np \). In our case, this turns out to be \( 20 \times 0.1 = 2 \), highlighting that, on average, we expect 2 people to respond when 20 are contacted.

Variance \( \text{Var}(X) \) measures how spread out the responses can be. It uses the formula \( \text{Var}(X) = np(1-p) \). For this exercise, that leads to \( 20 \times 0.1 \times 0.9 = 1.8 \), indicating moderate variability around the expected number of responses.

By understanding both expected value and variance, we get a fuller picture of our random variable, aiding in preparation for various outcomes and bettering decision-making processes in similar scenarios.