📊honors statistics review

key term - E(X)

Definition

E(X), or the expected value of a random variable X, represents the long-term average or mean of the possible values that X can take on. It is a measure of central tendency that describes the typical or central value of the probability distribution of X.

5 Must Know Facts For Your Next Test

  1. The expected value, E(X), is the weighted average of all possible values of the random variable X, where the weights are the respective probabilities of each value occurring.
  2. For a discrete random variable X with possible values $x_1, x_2, ..., x_n$ and corresponding probabilities $p_1, p_2, ..., p_n$, the expected value is calculated as: $E(X) = \sum_{i=1}^n x_i \cdot p_i$.
  3. For a continuous random variable X with a probability density function $f(x)$, the expected value is calculated as: $E(X) = \int_{-\infty}^{\infty} x \cdot f(x) \, dx$.
  4. The expected value, E(X), is a fundamental concept in probability and statistics, as it allows for the quantification of the central tendency or average behavior of a random variable.
  5. The expected value, E(X), is a linear operator, meaning that $E(aX + b) = a \cdot E(X) + b$, where a and b are constants.

Review Questions

  • Explain how the expected value, E(X), is calculated for a discrete random variable.
    • For a discrete random variable X with possible values $x_1, x_2, ..., x_n$ and corresponding probabilities $p_1, p_2, ..., p_n$, the expected value, E(X), is calculated as the weighted average of all possible values of X. Specifically, E(X) = $\sum_{i=1}^n x_i \cdot p_i$, where each possible value $x_i$ is multiplied by its associated probability $p_i$, and the products are summed to obtain the expected value. This calculation provides the long-term average or central tendency of the random variable X.
  • Describe the relationship between the expected value, E(X), and the variance of a random variable X.
    • The expected value, E(X), and the variance of a random variable X are closely related. While the expected value represents the central tendency or average behavior of X, the variance measures the spread or dispersion of the probability distribution of X. Specifically, the variance of X is defined as the average squared deviation of X from its expected value, E(X). This relationship highlights that the expected value provides information about the typical or central value of X, while the variance quantifies how much the values of X tend to deviate from this central tendency.
  • Explain how the properties of linearity and additivity apply to the expected value, E(X).
    • The expected value, E(X), exhibits the important properties of linearity and additivity. The linearity property states that for any constants a and b, E(aX + b) = a \cdot E(X) + b. This means that the expected value of a linear transformation of a random variable is the linear transformation of the expected value of the original variable. The additivity property states that for any two independent random variables X and Y, E(X + Y) = E(X) + E(Y). This allows the expected value of a sum of random variables to be calculated as the sum of their individual expected values. These properties make the expected value a powerful and versatile tool in probability and statistics.