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A random variable is a fundamental concept in probability and statistics. It represents a quantity whose possible values are numerical outcomes of a random phenomenon. Think of a random variable as a way to assign numbers to the outcomes of a random process. For example, if you roll a die, the number that comes up is a value of the random variable.
There are two types of random variables: discrete and continuous. Discrete random variables can take on a finite number of values, like the result of rolling a die. Continuous random variables can take on any value within a given range, such as the height of people.
In the context of this exercise, the random variable in question is denoted as \( Y \). It has a specific mean and variance, which are key to using Tchebysheff's Theorem effectively. Understanding random variables helps form the basis for analyzing expected outcomes and the variability of those outcomes.

The mean and variance are statistical measures that describe the characteristics of a random variable. The mean, often represented by \( \mu \), is the average value of all possible outcomes of the random variable. It tells us the central point around which the values of the random variable cluster.
The variance, denoted by \( \sigma^2 \), measures the spread of the random variable's values around the mean. It is the average of the squared differences from the mean. The larger the variance, the more spread out the values are.
In this exercise, the random variable \( Y \) has a mean of 11 and a variance of 9. These parameters are crucial as they guide us in applying Tchebysheff's Theorem. Specifically, knowing the variance allows us to compute the standard deviation, which is the square root of the variance, essential for further steps in the problem.

Probability bounds refer to the limits within which the outcome of a random variable is expected to lie, with a certain level of confidence. Tchebysheff's Theorem provides a way to calculate these bounds by relating them to the mean and standard deviation of the variable.
With Tchebysheff's Theorem, we can determine how likely it is for a random variable to fall within a specific interval around its mean, using the formula \[ P(|Y - \mu| < k\sigma) \geq 1 - \frac{1}{k^2} \]This theorem is powerful because it applies to any random distribution as long as we know the mean and variance.
In our problem, we use the theorem to find a lower bound for the probability that \( Y \) is between 6 and 16. Using the calculated \( k = \frac{5}{3} \), we figured out that the probability is at least 0.64. This means we can assert with at least 64% confidence that \( Y \) falls within that range.

The standard deviation is a measure of the amount of variation or dispersion of a set of values. It is the square root of the variance and provides a metric for how much a random variable deviates from its mean on average. Generally, a smaller standard deviation means the values are tightly clustered around the mean, while a larger standard deviation indicates the values are more spread out.
In mathematical terms, if the variance of \( Y \) is 9, then the standard deviation \( \sigma \) is calculated as:\[ \sigma = \sqrt{9} = 3 \]This is a crucial step because the standard deviation directly influences the calculation of \( k \), the number of standard deviations used in Tchebysheff's Theorem.
For instance, when determining the value of \( C \) such that \( P(|Y-11| \geq C) \leq 0.09 \), we calculated \( k \) and then applied it to find that \( C = 10 \). The standard deviation helps scale \( k \) into the actual distance in the original units of \( Y \).